2.3: The span of a set of vectors
\(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\twovec}[2]{\begin{pmatrix} #1 \\ #2 \end{pmatrix} } \) \(\newcommand{\threevec}[3]{\begin{pmatrix} #1 \\ #2 \\ #3 \end{pmatrix} } \) \(\newcommand{\fourvec}[4]{\begin{pmatrix} #1 \\ #2 \\ #3 \\ #4 \end{pmatrix} } \) \(\newcommand{\fivevec}[5]{\begin{pmatrix} #1 \\ #2 \\ #3 \\ #4 \\ #5 \end{pmatrix} } \)Our work in this chapter enables us to rewrite a linear system in the form \(A\mathbf x = \mathbf b\text{.}\) Besides being a more compact way of expressing a linear system, this form allows us to think about linear systems geometrically since matrix multiplication is defined in terms of linear combinations of vectors.
We now return, in this and the next section, to the two fundamental questions asked in Question 1.4.2.
- Existence: Is there a solution to the equation \(A\mathbf x=\mathbf b\text{?}\)
- Uniqueness: If there is a solution to the equation \(A\mathbf x=\mathbf b\text{,}\) is it unique?
In this section, we focus on the existence question and introduce the concept of span to provide a framework for thinking about it geometrically.
Preview Activity 2.3.1. The existence of solutions.
- If the equation \(A\mathbf x = \mathbf b\) is inconsistent, what can we say about the pivots of the augmented matrix \(\left[\begin{array}{r|r} A & \mathbf b \end{array}\right]\text{?}\)
-
Consider the matrix \(A\)
\begin{equation*} A = \left[ \begin{array}{rrr} 1 & 0 & -2 \\ -2 & 2 & 2 \\ 1 & 1 & -3 \end{array}\right]\text{.} \end{equation*}
If \(\mathbf b=\threevec{2}{2}{5}\text{,}\) is the equation \(A\mathbf x = \mathbf b\) consistent? If so, find a solution.
-
If \(\mathbf b=\threevec{2}{2}{6}\text{,}\) is the equation \(A\mathbf x = \mathbf b\) consistent? If so, find a solution.
- Identify the pivot positions of \(A\text{.}\)
- For our two choices of the vector \(\mathbf b\text{,}\) one equation \(A\mathbf x = \mathbf b\) has a solution and the other does not. What feature of the pivot positions of the matrix \(A\) tells us to expect this?
2.3.1 The span of a set of vectors
In the preview activity, we considered a \(3\times3\) matrix \(A\) and found that the equation \(A\mathbf x = \mathbf b\) has a solution for some vectors \(\mathbf b\) in \(\mathbb R^3\) and has no solution for others. We will introduce a concept called span that describes the vectors \(\mathbf b\) for which there is a solution.
Since we would like to think about this concept geometrically, we will consider an \(m\times n\) matrix \(A\) as being composed of \(n\) vectors in \(\mathbb R^m\text{;}\) that is,
Remember that Proposition 2.2.4 says that the equation \(A\mathbf x = \mathbf b\) is consistent if and only if we can express \(\mathbf b\) as a linear combination of \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\text{.}\)
The span of a set of vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) is the set of all linear combinations of the vectors.
In other words, the span of \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) consists of all the vectors \(\mathbf b\) for which the equation
is consistent.
The span of a set of vectors has an appealing geometric interpretation. Remember that we may think of a linear combination as a recipe for walking in \(\mathbb R^m\text{.}\) We first move a prescribed amount in the direction of \(\mathbf v_1\text{,}\) then a prescribed amount in the direction of \(\mathbf v_2\text{,}\) and so on. As the following activity will show, the span consists of all the places we can walk to.
Activity 2.3.2.
Let's look at two examples to develop some intuition for the concept of span.
a. First, we will consider the set of vectors
The diagram below can be used to construct linear combinations whose weights \(a\) and \(b\) may be varied using the sliders at the top. The vectors \({\mathbf v}\) and \({\mathbf w}\) are drawn in gray while the linear combination $$ a{\mathbf v} + b{\mathbf w} $$ is in red.
-
What vector is the linear combination of \(\mathbf v\) and \(\mathbf w\) with weights:
- \(a = 2\) and \(b=0\text{?}\)
- \(a = 1\) and \(b=1\text{?}\)
- \(a = 0\) and \(b=-1\text{?}\)
2. Can the vector \(\twovec{2}{4}\) be expressed as a linear combination of \(\mathbf v\) and \(\mathbf w\text{?}\) Is the vector \(\twovec{2}{4}\) in the span of \(\mathbf v\) and \(\mathbf w\text{?}\)
3. Can the vector \(\twovec{3}{0}\) be expressed as a linear combination of \(\mathbf v\) and \(\mathbf w\text{?}\) Is the vector \(\twovec{3}{0}\) in the span of \(\mathbf v\) and \(\mathbf w\text{?}\)
4. Describe the set of vectors in the span of \(\mathbf v\) and \(\mathbf w\text{.}\)
5. For what vectors \(\mathbf b\) does the equation
b. We will now look at an example where
In a similar way, the diagram below can be used to construct linear combinations \( a{\mathbf v} + b{\mathbf w}\text{.} \)
1. What vector is the linear combination of \(\ mathbf v\) and \(\ mathbf w\) with weights:
- \(a = 2\) and \(b=0\text{?}\)
- \(a = 1\) and \(b=1\text{?}\
- \(a = 0\) and \(b=-1\text{?}\)
2. Can the vector \(\twovec{-2}{2}\) be expressed as a linear combination of \(\mathbf v\) and \(\mathbf w\text{?}\) Is the vector \(\twovec{-2}{2}\) in the span of \(\mathbf v\) and \(\mathbf w\text{?}\)
3. Can the vector \(\twovec{3}{0}\) be expressed as a linear combination of \(\mathbf v\) and \(\mathbf w\text{?}\) Is the vector \(\twovec{3}{0}\) in the span of \(\mathbf v\) and \(\mathbf w\text{?}\)
4. Describe the set of vectors in the span of \(\mathbf v\) and \(\mathbf w\text{.}\)
5. For what vectors \(\mathbf b\) does the equation
have a solution?
Let's consider the first example in the previous activity. Here, the vectors \(\mathbf v\) and \(\mathbf w\) are scalar multiples of one another, which means that they lie on the same line. When we form linear combinations, we are allowed to walk only in the direction of \(\mathbf v\) and \(\mathbf w\text{,}\) which means we are constrained to stay on this same line. Therefore, the span of \(\mathbf v\) and \(\mathbf w\) consists only of this line.
With this choice of vectors \(\mathbf v\) and \(\mathbf w\text{,}\) all linear combinations lie on the line shown. This line, therefore, is the span of the vectors \(\mathbf v\) and \(\mathbf w\text{.}\)
We may see this algebraically since the vector \(\mathbf w = -2\mathbf v\text{.}\) Consequently, when we form a linear combination of \(\mathbf v\) and \(\mathbf w\text{,}\) we see that
Therefore, any linear combination of \(\mathbf v\) and \(\mathbf w\) reduces to a scalar multiple of \(\mathbf v\text{,}\) and we have seen that the scalar multiples of a nonzero vector form a line.
In the second example, however, the vectors are not scalar multiples of one another, and we see that we can construct any vector in \(\mathbb R^2\) as a linear combination of \(\mathbf v\) and \(\mathbf w\text{.}\)
With this choice of vectors \(\mathbf v\) and \(\mathbf w\text{,}\) we are able to form any vector in \(\mathbb R^2\) as a linear combination. Therefore, the span of the vectors \(\mathbf v\) and \(\mathbf w\) is the entire plane, \(\mathbb R^2\text{.}\)
Once again, we can see this algebraically. Asking if the vector \(\mathbf b\) is in the span of \(\mathbf v\) and \(\mathbf w\) is the same as asking if the linear system
is consistent.
The augmented matrix for this system is
Since it is impossible to obtain a pivot in the rightmost column, we know that this system is consistent no matter what the vector \(\mathbf b\) is. Therefore, every vector \(\mathbf b\) in \(\mathbb R^2\) is in the span of \(\mathbf v\) and \(\mathbf w\text{.}\)
In this case, notice that the reduced row echelon form of the matrix
has a pivot in every row. When this happens, it is not possible for any augmented matrix to have a pivot in the rightmost column. Therefore, the linear system is consistent for every vector \(\mathbf b\text{,}\) which implies that the span of \(\mathbf v\) and \(\mathbf w\) is \(\mathbb R^2\text{.}\)
Notation 2.3.4.
We will denote the span of the set of vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) by \(Span \{{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\} \text{.}\)
2.3.2 Pivot positions and span
In the previous activity, we saw two examples, both of which considered two vectors \(\mathbf v\) and \(\mathbf w\) in \(\mathbb R^2\text{.}\) In one example, the \(Span \{{\mathbf v,\mathbf w}\} \) consisted of a line; in the other, the \(Span \{{\mathbf v,\mathbf w}=\mathbb R^2\} \text{.}\) We would like to be able to distinguish these two situations in a more algebraic fashion. After all, we will need to be able to deal with vectors in many more dimensions where we will not be able to draw pictures.
The key is found by looking at the pivot positions of the matrix \(\left[\begin{array}{rrrr} \mathbf v_1& \mathbf v_2& \ldots\mathbf v_n \end{array}\right] \text{.}\) In the first example, the matrix whose columns are \(\mathbf v\) and \(\mathbf w\) is
which has exactly one pivot position. We found the \(Span \{{\mathbf v,\mathbf w}\} \) to be a line, in this case.
In the second example, this matrix is
which has two pivot positions. Here, we found \(Span \{{\mathbf v,\mathbf w}\} =\mathbb R^2\text{.}\)
These examples point to the fact that the size of the span is related to the number of pivot positions. We will develop this idea more fully in Section 2.4 and Section 3.5 . For now, however, we will examine the possibilities in \(\mathbb R^3\text{.}\)
Activity 2.3.3.
In this activity, we will look at the span of sets of vectors in \(\mathbb R^3\text{.}\)
-
Suppose \(v=\threevec{1}{2}{1}\text{.}\) Give a written description of \(Span \{{v}\} \) and a rough sketch of it below.
-
Consider now the two vectors
\begin{equation*} \mathbf e_1 = \threevec{1}{0}{0}, \mathbf e_2 = \threevec{0}{1}{0}\text{.} \end{equation*}
Sketch the vectors below. Then give a written description of \(Span \{{\mathbf e_1,\mathbf e_2}\} \) and a rough sketch of it below.
Let's now look at this algebraically by writing write \(\mathbf b = \threevec{b_1}{b_2}{b_3}\text{.}\) Determine the conditions on \(b_1\text{,}\) \(b_2\text{,}\) and \(b_3\) so that \(\mathbf b\) is in \(Span \{{\mathbf e_1,\mathbf e_2}\} \) by considering the linear system
\begin{equation*} \left[\begin{array}{rr} \mathbf e_1 & \mathbf e_2 \\ \end{array}\right] \mathbf x = \mathbf b \end{equation*}or
\begin{equation*} \left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ \end{array}\right] \mathbf x = \threevec{b_1}{b_2}{b_3}\text{.} \end{equation*}Explain how this relates to your sketch of \(Span \{{\mathbf e_1,\mathbf e_2}\} \text{.}\)
-
Consider the vectors
\begin{equation*} \mathbf v_1 = \threevec{1}{1}{-1}, \mathbf v_2 = \threevec{0}{2}{1}\text{.} \end{equation*}
-
Is the vector \(\mathbf b=\threevec{1}{-2}{4}\) in \(Span \{{\mathbf v_1,\mathbf v_2}\} \text{?}\)
-
Is the vector \(\mathbf b=\threevec{-2}{0}{3}\) in \(Span \{{\mathbf v_1,\mathbf v_2}\} \text{?}\)
- Give a written description of \(Span \{{\mathbf v_1,\mathbf v_2}\} \text{.}\)
-
Is the vector \(\mathbf b=\threevec{1}{-2}{4}\) in \(Span \{{\mathbf v_1,\mathbf v_2}\} \text{?}\)
-
Consider the vectors
\begin{equation*} \mathbf v_1 = \threevec{1}{1}{-1}, \mathbf v_2 = \threevec{0}{2}{1}, \mathbf v_3 = \threevec{1}{-2}{4}\text{.} \end{equation*}
Form the matrix \(\left[\begin{array}{rrrr} \mathbf v_1 & \mathbf v_2 & \mathbf v_3 \end{array}\right]\) and find its reduced row echelon form.
What does this tell you about \(Span \{{\mathbf v_1,\mathbf v_2,\mathbf v_3}\} \text{?}\) - If a set of vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) spans \(\mathbb R^3\text{,}\) what can you say about the pivots of the matrix \(\left[\begin{array}{rrrr} \mathbf v_1& \mathbf v_2& \ldots& \mathbf v_n \end{array}\right]\text{?}\)
- What is the smallest number of vectors such that \(Span \{{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\} = \mathbb R^3\text{?}\)
This activity shows us the types of sets that can appear as the span of a set of vectors in \(\mathbb R^3\text{.}\)
-
First, with a single vector, all linear combinations are simply scalar multiples of that vector, which creates a line.
Notice that the matrix formed by this vector has one pivot, just as in our earlier example in \(\mathbb R^2\text{.}\)
\begin{equation*} \threevec{1}{2}{1} \sim \threevec{1}{0}{0}\text{.} \end{equation*}
-
When we consider linear combinations of the vectors
\begin{equation*} \mathbf e_1 = \threevec{1}{0}{0}, \mathbf e_2 = \threevec{0}{1}{0}\text{,} \end{equation*}
we must obtain vectors of the form
\begin{equation*} a\mathbf e_1 + b\mathbf e_2 = a\threevec{1}{0}{0}+b\threevec{0}{1}{0} = \threevec{a}{b}{0}\text{.} \end{equation*}Since the third component is zero, these vectors form the plane \(z=0\text{.}\)
-
Notice here that the matrix composed of the vectors has two pivot positions.
\begin{equation*} \left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ \end{array}\right]\text{.} \end{equation*}
-
Similarly, the span of the vectors
\begin{equation*} \mathbf v_1 = \threevec{1}{1}{-1}, \mathbf v_2 = \threevec{0}{2}{1}\text{,} \end{equation*}
will form a plane.
We saw one vector \(\mathbf b\) that was not in \(Span \{{\mathbf v_1,\mathbf v_2}\} \) and one that is.Once again, the matrix\begin{equation*} \left[\begin{array}{rr} \mathbf v_1 & \mathbf v_2 \end{array}\right] = \left[\begin{array}{rr} 1 & 0 \\ 1 & 2 \\ -1 & 1 \\ \end{array}\right] \sim \left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ \end{array}\right] \end{equation*}has two pivot positions.
-
Finally, we looked at a set of vectors whose matrix
\begin{equation*} \left[\begin{array}{rrr} \mathbf v_1 & \mathbf v_2 & \mathbf v_3 \end{array}\right] = \left[\begin{array}{rrr} 1 & 0 & 1 \\ 1 & 2 & -2 \\ -1 & 1 & 4 \\ \end{array}\right] \sim \left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}\right] \end{equation*}
has three pivot positions, one in every row. This is significant because it means that if we consider an augmented matrix
\begin{equation*} \left[\begin{array}{rrr|r} 1 & 0 & 1 & *\\ 1 & 2 & -2 & * \\ -1 & 1 & 4 & * \\ \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & *\\ 0 & 1 & 0 & * \\ 0 & 0 & 1 & * \\ \end{array}\right]\text{,} \end{equation*}there cannot be a pivot position in the rightmost column. This linear system is consistent for every vector \(\mathbf b\text{,}\) which tells us that \(Span \{{\mathbf v_1,\mathbf v_2,\mathbf v_3}\} = \mathbb R^3\text{.}\)
To summarize, we looked at the pivot positions in the matrix whose columns were the vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\text{.}\) We found that with
- one pivot position, the span was a line.
- two pivot positions, the span was a plane.
- three pivot positions, the span was \(\mathbb R^3\text{.}\)
Once again, we will develop these ideas more fully in the next and subsequent sections. However, we saw that, when considering vectors in \(\mathbb R^3\text{,}\) a pivot position in every row implied that the span of the vectors is \(\mathbb R^3\text{.}\) The same reasoning applies more generally.
Suppose we have vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) in \(\mathbb R^m\text{.}\) Then \(Span \{{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\} =\mathbb R^m\) if and only if the matrix \(\left[\begin{array}{rrrr} \mathbf v_1& \mathbf v_2& \ldots& \mathbf v_n \end{array}\right]\) has a pivot position in every row.
This tells us something important about the number of vectors needed to span \(\mathbb R^m\text{.}\) Suppose we have \(n\) vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) that span \(\mathbb R^m\text{.}\) The proposition tells us that the matrix \(A = \left[\begin{array}{rrrr} \mathbf v_1& \mathbf v_2\ldots\mathbf v_n \end{array}\right]\) has a pivot position in every row, such as in this reduced row echelon matrix.
Since a matrix can have at most one pivot position in a column, there must be at least as many columns as there are rows, which implies that \(n\geq m\text{.}\)
For instance, if we have a set of vectors that span \(\mathbb R^{632}\text{,}\) there must be at least 632 vectors in the set.
If a set of vectors span \(\mathbb R^m\text{,}\) there must be at least \(m\) vectors in the set.
This makes sense intuitively. We have thought about a linear combination of a set of vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) as the result of walking a certain distance in the direction of \(\mathbf v_1\text{,}\) followed by walking a certain distance in the direction of \(\mathbf v_2\text{,}\) and so on. If \(Span \{{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\} = \mathbb R^m\text{,}\) this means that we can walk to any point in \(\mathbb R^m\) using the directions \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\text{.}\) It makes sense that we would need at least \(m\) directions to give us the flexibilty needed to reach any point in \(\mathbb R^m\text{.}\)
Because span is a concept that is connected to a set of vectors, we say, "The span of the set of vectors \(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\) is ...." While it may be tempting to say, "The span of the matrix \(A\) is ...," we should instead say "The span of the columns of the matrix \(A\) is ...."
Summary
We defined the span of a set of vectors and developed some intuition for this concept through a series of examples.
- The span of a set of vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) is the set of linear combinations of the vectors. We denote the span by \(Span \{{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\} \text{.}\)
-
A vector \(\mathbf b\) is in \(Span \{{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\} \) if an only if the linear system
\begin{equation*} \left[\begin{array}{rrrr} \mathbf v_1& \mathbf v_2& \ldots\mathbf v_n \end{array}\right] \mathbf x = \mathbf b \end{equation*}
is consistent.
-
If the \(m\times n\) matrix
\begin{equation*} \left[\begin{array}{rrrr} \mathbf v_1& \mathbf v_2& \ldots\mathbf v_n \end{array}\right] \end{equation*}
has a pivot in every row, then the span of these vectors is \(\mathbb R^m\text{;}\) that is, \(Span \{{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\} = \mathbb R^m\text{.}\)
- Any set of vectors that spans \(\mathbb R^m\) must have at least \(m\) vectors.
Exercises 2.3.4Exercises
In this exercise, we will consider the span of some sets of two- and three-dimensional vectors.
-
Consider the vectors
\begin{equation*} \mathbf v_1 = \twovec{1}{-2}, \mathbf v_2 = \twovec{4}{3}\text{.} \end{equation*}
- Is \(\mathbf b = \twovec{2}{1}\) in \(Span \{{\mathbf v_1,\mathbf v_2}\} \text{?}\)
- Give a written description of \(Span \{{\mathbf v_1,\mathbf v_2}\} \text{.}\)
-
Consider the vectors
\begin{equation*} \mathbf v_1=\threevec{2}{1}{3}, \mathbf v_2=\threevec{-2}{0}{2}, \mathbf v_3=\threevec{6}{1}{-1}\text{.} \end{equation*}
- Is the vector \(\mathbf b=\threevec{-10}{-1}{5}\) in \(Span \{{\mathbf v_1,\mathbf v_2,\mathbf v_3}\} \text{?}\)
- Is the vector \(\mathbf v_3\) in \(Span \{{\mathbf v_1,\mathbf v_2,\mathbf v_3}\} \text{?}\)
- Is the vector \(\mathbf b=\threevec{3}{3}{-1}\) in \(Span \{{\mathbf v_1,\mathbf v_2,\mathbf v_3}\} \text{?}\)
- Give a written description of \(Span \{{\mathbf v_1,\mathbf v_2,\mathbf v_3}\} \text{.}\)
Provide a justification for your response to the following questions.
- Suppose you have a set of vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\text{.}\) Can you guarantee that \(\zerovec\) is in \(Span \{{\mathbf v_1\,\mathbf v_2,\ldots,\mathbf v_n}\} \text{?}\)
- Suppose that \(A\) is an \(m \times n\) matrix. Can you guarantee that the equation \(A\mathbf x = \zerovec\) is consistent?
- What is \(Span \{{\zerovec,\zerovec,\ldots,\zerovec}\} \text{?}\)
For both parts of this exericse, give a written description of sets of the vectors \(\mathbf b\) and include a sketch.
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For which vectors \(\mathbf b\) in \(\mathbb R^2\) is the equation
\begin{equation*} \left[\begin{array}{rr} 3 & -6 \\ -2 & 4 \\ \end{array}\right] \mathbf x = \mathbf b \end{equation*}
consistent?
-
For which vectors \(\mathbf b\) in \(\mathbb R^2\) is the equation
\begin{equation*} \left[\begin{array}{rr} 3 & -6 \\ -2 & 2 \\ \end{array}\right] \mathbf x = \mathbf b \end{equation*}
consistent?
Consider the following matrices:
Do the columns of \(A\) span \(\mathbb R^4\text{?}\) Do the columns of \(B\) span \(\mathbb R^4\text{?}\)
Determine whether the following statements are true or false and provide a justification for your response. Throughout, we will assume that the matrix \(A\) has columns \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\text{;}\) that is,
- If the equation \(A\mathbf x = \mathbf b\) is consistent, then \(\mathbf b\) is in \(Span \{{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\} \text{.}\)
- The equation \(A\mathbf x = \mathbf v_1\) is always consistent.
- If \(\mathbf v_1\text{,}\) \(\mathbf v_2\text{,}\) \(\mathbf v_3\text{,}\) and \(\mathbf v_4\) are vectors in \(\mathbb R^3\text{,}\) then their span is \(\mathbb R^3\text{.}\)
- If \(\mathbf b\) can be expressed as a linear combination of \(\mathbf v_1, \mathbf v_2,\ldots,\mathbf v_n\text{,}\) then \(\mathbf b\) is in \(Span \{{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\} \text{.}\)
- If \(A\) is a \(8032\times 427\) matrix, then the span of the columns of \(A\) is a set of vectors in \(\mathbb R^{427}\text{.}\)
This exercise asks you to construct some matrices whose columns span a given set.
- Construct a \(3\times3\) matrix whose columns span \(\mathbb R^3\text{.}\)
- Construct a \(3\times3\) matrix whose columns span a plane in \(\mathbb R^3\text{.}\)
- Construct a \(3\times3\) matrix whose columns span a line in \(\mathbb R^3\text{.}\)
Provide a justification for your response to the following questions.
- Suppose that we have vectors in \(\mathbb R^8\text{,}\) \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_{10}\text{,}\) whose span is \(\mathbb R^8\text{.}\) Can every vector \(\mathbf b\) in \(\mathbb R^8\) be written as a linear combination of \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_{10}\text{?}\)
- Suppose that we have vectors in \(\mathbb R^8\text{,}\) \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_{10}\text{,}\) whose span is \(\mathbb R^8\text{.}\) Can every vector \(\mathbf b\) in \(\mathbb R^8\) be written uniquely as a linear combination of \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_{10}\text{?}\)
-
Do the vectors
\begin{equation*} \mathbf e_1=\threevec{1}{0}{0}, \mathbf e_2=\threevec{0}{1}{0}, \mathbf e_3=\threevec{0}{0}{1} \end{equation*}
span \(\mathbb R^3\text{?}\)
- Suppose that \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) span \(\mathbb R^{438}\text{.}\) What can you guarantee about the value of \(n\text{?}\)
- Can 17 vectors in \(\mathbb R^{20}\) span \(\mathbb R^{20}\text{?}\)
The following observation will be helpful in this exericse. If we want to find a solution to the equation \(AB\mathbf x = \mathbf b\text{,}\) we could first find a solution to the equation \(A\mathbf y = \mathbf b\) and then find a solution to the equation \(B\mathbf x = \mathbf y\text{.}\)
Suppose that \(A\) is a \(3\times 4\) matrix whose columns span \(\mathbb R^3\) and \(B\) is a \(4\times 5\) matrix. In this case, we can form the product \(AB\text{.}\)
- What are the dimensions of the product \(AB\text{?}\)
- Can you guarantee that the columns of \(AB\) span \(\mathbb R^3\text{?}\)
- If you know additionally that the span of the columns of \(B\) is \(\mathbb R^4\text{,}\) can you guarantee that the columns of \(AB\) span \(\mathbb R^3\text{?}\)
Suppose that \(A\) is a \(12\times12\) matrix and that, for some vector \(\mathbf b\text{,}\) the equation \(A\mathbf x=\mathbf b\) has a unique solution.
- What can you say about the pivot positions of \(A\text{?}\)
- What can you say about the span of the columns of \(A\text{?}\)
- If \(\mathbf c\) is some other vector in \(\mathbb R^{12}\text{,}\) what can you conclude about the equation \(A\mathbf x = \mathbf c\text{?}\)
- What can you about the solution space to the equation \(A\mathbf x =\zerovec\text{?}\)
Suppose that
- Is \(\mathbf v_3\) a linear combination of \(\mathbf v_1\) and \(\mathbf v_2\text{?}\) If so, find weights such that \(\mathbf v_3 = a\mathbf v_1+b\mathbf v_2\text{.}\)
-
Show that a linear combination
\begin{equation*} a\mathbf v_1 + b\mathbf v_2 + c\mathbf v_3 \end{equation*}
can be rewritten as a linear combination of \(\mathbf v_1\) and \(\mathbf v_2\text{.}\)
- Explain why \(Span \{{\mathbf v_1,\mathbf v_2,\mathbf v_3}\} = Span \{{\mathbf v_1,\mathbf v_2}\} \text{.}\)
As defined in this section, the span of a set of vectors is generated by taking all possible linear combinations of those vectors. This exericse will demonstrate the fact that the span can also be realized as the solution space to a linear system.
We will consider the vectors
-
Is every vector in \(\mathbb R^3\) in \(Span \{{\mathbf v_1,\mathbf v_2,\mathbf v_3}\} \text{?}\) If not, describe the span.
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To describe \(Span \{{\mathbf v_1,\mathbf v_2,\mathbf v_3}\} \) as the solution space of a linear system, we will write
\begin{equation*} \mathbf b=\threevec{a}{b}{c}\text{.} \end{equation*}
If \(\mathbf b\) is in \(Span \{{\mathbf v_1,\mathbf v_2,\mathbf v_3}\} \text{,}\) then the linear system corresponding to the augmented matrix
\begin{equation*} \left[\begin{array}{rrr|r} 1 & 2 & 1 & a \\ 0 & 1 & 1 & b \\ -2& 0 & 2 & c \\ \end{array}\right] \end{equation*}must be consistent. This means that a pivot cannot occur in the rightmost column. Perform row operations to put this augmented matrix into a triangular form. Now identify an equation in \(a\text{,}\) \(b\text{,}\) and \(c\) that tells us when there is no pivot in the rightmost column. The solution space to this equation describes \(Span \{{\mathbf v_1,\mathbf v_2,\mathbf v_3}\} \text{.}\)
- In this example, the matrix formed by the vectors \(\left[\begin{array}{rrr} \mathbf v_1& \mathbf v_2& \mathbf v_2 \\ \end{array}\right]\) has two pivot positions. Suppose we were to consider another example in which this matrix had had only one pivot position. How would this have changed the linear system describing \(Span \{{\mathbf v_1,\mathbf v_2,\mathbf v_3}\} \text{?}\)


3.3.0: Bond Types
Learning Objectives
- Explain the formation of cations, anions, and ionic compounds
- Predict the charge of common metallic and nonmetallic elements, and write their electron configurations
- Describe the formation of covalent bonds
- Define electronegativity and assess the polarity of covalent bonds
Video \(\PageIndex{1}\) : Why do atoms bond?
Video \(\PageIndex{2}\) : And how do atoms bond?
As you have learned, ions are atoms or molecules bearing an electrical charge. A cation (a positive ion) forms when a neutral atom loses one or more electrons from its valence shell, and an anion (a negative ion) forms when a neutral atom gains one or more electrons in its valence shell.
Compounds composed of ions are called ionic compounds (or salts), and their constituent ions are held together by ionic bonds : electrostatic forces of attraction between oppositely charged cations and anions. The properties of ionic compounds shed some light on the nature of ionic bonds. Ionic solids exhibit a crystalline structure and tend to be rigid and brittle; they also tend to have high melting and boiling points, which suggests that ionic bonds are very strong. Ionic solids are also poor conductors of electricity for the same reason—the strength of ionic bonds prevents ions from moving freely in the solid state. Most ionic solids, however, dissolve readily in water. Once dissolved or melted, ionic compounds are excellent conductors of electricity and heat because the ions can move about freely.
Neutral atoms and their associated ions have very different physical and chemical properties. Sodium atoms form sodium metal, a soft, silvery-white metal that burns vigorously in air and reacts explosively with water. Chlorine atoms form chlorine gas, Cl 2 , a yellow-green gas that is extremely corrosive to most metals and very poisonous to animals and plants. The vigorous reaction between the elements sodium and chlorine forms the white, crystalline compound sodium chloride, common table salt, which contains sodium cations and chloride anions (Figure Figure \(\PageIndex{1}\)). The compound composed of these ions exhibits properties entirely different from the properties of the elements sodium and chlorine. Chlorine is poisonous, but sodium chloride is essential to life; sodium atoms react vigorously with water, but sodium chloride simply dissolves in water.
Figure \(\PageIndex{1}\) : (a) Sodium is a soft metal that must be stored in mineral oil to prevent reaction with air or water. (b) Chlorine is a pale yellow-green gas. (c) When combined, they form white crystals of sodium chloride (table salt). (credit a: modification of work by “Jurii”/Wikimedia Commons)
The Formation of Ionic Compounds
Binary ionic compounds are composed of just two elements: a metal (which forms the cations) and a nonmetal (which forms the anions). For example, NaCl is a binary ionic compound. We can think about the formation of such compounds in terms of the periodic properties of the elements. Many metallic elements have relatively low ionization potentials and lose electrons easily. These elements lie to the left in a period or near the bottom of a group on the periodic table. Nonmetal atoms have relatively high electron affinities and thus readily gain electrons lost by metal atoms, thereby filling their valence shells. Nonmetallic elements are found in the upper-right corner of the periodic table.
As all substances must be electrically neutral, the total number of positive charges on the cations of an ionic compound must equal the total number of negative charges on its anions. The formula of an ionic compound represents the simplest ratio of the numbers of ions necessary to give identical numbers of positive and negative charges. For example, the formula for aluminum oxide, Al 2 O 3 , indicates that this ionic compound contains two aluminum cations, Al 3+ , for every three oxide anions, O 2− [thus, (2 × +3) + (3 × –2) = 0].
It is important to note, however, that the formula for an ionic compound does not represent the physical arrangement of its ions. It is incorrect to refer to a sodium chloride (NaCl) “molecule” because there is not a single ionic bond, per se, between any specific pair of sodium and chloride ions. The attractive forces between ions are isotropic—the same in all directions—meaning that any particular ion is equally attracted to all of the nearby ions of opposite charge. This results in the ions arranging themselves into a tightly bound, three-dimensional lattice structure. Sodium chloride, for example, consists of a regular arrangement of equal numbers of Na + cations and Cl – anions (Figure Figure \(\PageIndex{2}\)).
Figure \(\PageIndex{2}\) : The atoms in sodium chloride (common table salt) are arranged to (a) maximize opposite charges interacting. The smaller spheres represent sodium ions, the larger ones represent chloride ions. In the expanded view (b), the geometry can be seen more clearly. Note that each ion is “bonded” to all of the surrounding ions—six in this case.
The strong electrostatic attraction between Na + and Cl – ions holds them tightly together in solid NaCl. It requires 769 kJ of energy to dissociate one mole of solid NaCl into separate gaseous Na + and Cl – ions:
Video \(\PageIndex{3}\) : A review of ionic bonds from last term.
Electronic Structures of Cations
When forming a cation, an atom of a main group element tends to lose all of its valence electrons, thus assuming the electronic structure of the noble gas that precedes it in the periodic table. For groups 1 (the alkali metals) and 2 (the alkaline earth metals), the group numbers are equal to the numbers of valence shell electrons and, consequently, to the charges of the cations formed from atoms of these elements when all valence shell electrons are removed. For example, calcium is a group 2 element whose neutral atoms have 20 electrons and a ground state electron configuration of 1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 4 s 2 . When a Ca atom loses both of its valence electrons, the result is a cation with 18 electrons, a 2+ charge, and an electron configuration of 1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 . The Ca 2+ ion is therefore isoelectronic with the noble gas Ar.
For groups 12–17, the group numbers exceed the number of valence electrons by 10 (accounting for the possibility of full d subshells in atoms of elements in the fourth and greater periods). Thus, the charge of a cation formed by the loss of all valence electrons is equal to the group number minus 10. For example, aluminum (in group 13) forms 3+ ions (Al 3+ ).
Exceptions to the expected behavior involve elements toward the bottom of the groups. In addition to the expected ions Tl 3+ , Sn 4+ , Pb 4+ , and Bi 5+ , a partial loss of these atoms’ valence shell electrons can also lead to the formation of Tl + , Sn 2+ , Pb 2+ , and Bi 3+ ions. The formation of these 1+, 2+, and 3+ cations is ascribed to the inert pair effect , which reflects the relatively low energy of the valence s -electron pair for atoms of the heavy elements of groups 13, 14, and 15. Mercury (group 12) also exhibits an unexpected behavior: it forms a diatomic ion, \(\ce{Hg_2^2+}\) (an ion formed from two mercury atoms, with an Hg-Hg bond), in addition to the expected monatomic ion Hg 2+ (formed from only one mercury atom).
Transition and inner transition metal elements behave differently than main group elements. Most transition metal cations have 2+ or 3+ charges that result from the loss of their outermost s electron(s) first, sometimes followed by the loss of one or two d electrons from the next-to-outermost shell. For example, iron (1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 3 d 6 4 s 2 ) forms the ion Fe 2+ (1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 3 d 6 ) by the loss of the 4 s electrons and the ion Fe 3+ (1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 3 d 5 ) by the loss of the 4 s electrons and one of the 3 d electrons. Although the d orbitals of the transition elements are—according to the Aufbau principle—the last to fill when building up electron configurations, the outermost s electrons are the first to be lost when these atoms ionize. When the inner transition metals form ions, they usually have a 3+ charge, resulting from the loss of their outermost s electrons and a d or f electron.
Example \(\PageIndex{1}\): Determining the Electronic Structures of Cations
There are at least 14 elements categorized as “essential trace elements” for the human body. They are called “essential” because they are required for healthy bodily functions, “trace” because they are required only in small amounts, and “elements” in spite of the fact that they are really ions. Two of these essential trace elements, chromium and zinc, are required as Cr 3+ and Zn 2+ . Write the electron configurations of these cations.
Solution
First, write the electron configuration for the neutral atoms:
- Zn: [Ar]3 d 10 4 s 2
- Cr: [Ar]3 d 5 4 s 1
Next, remove electrons from the highest energy orbital. For the transition metals, electrons are removed from the s orbital first and then from the d orbital. For the p -block elements, electrons are removed from the p orbitals and then from the s orbital. Zinc is a member of group 12, so it should have a charge of 2+, and thus loses only the two electrons in its s orbital. Chromium is a transition element and should lose its s electrons and then its d electrons when forming a cation. Thus, we find the following electron configurations of the ions:
- Zn 2+ : [Ar]3 d 10
- Cr 3+ : [Ar]3 d 3
Exercise \(\PageIndex{1}\)
Potassium and magnesium are required in our diet. Write the electron configurations of the ions expected from these elements.
- Answer
-
K + : [Ar], Mg 2+ : [Ne]
Electronic Structures of Anions
Most monatomic anions form when a neutral nonmetal atom gains enough electrons to completely fill its outer s and p orbitals, thereby reaching the electron configuration of the next noble gas. Thus, it is simple to determine the charge on such a negative ion: The charge is equal to the number of electrons that must be gained to fill the s and p orbitals of the parent atom. Oxygen, for example, has the electron configuration 1 s 2 2 s 2 2 p 4 , whereas the oxygen anion has the electron configuration of the noble gas neon (Ne), 1 s 2 2 s 2 2 p 6 . The two additional electrons required to fill the valence orbitals give the oxide ion the charge of 2– (O 2– ).
Example \(\PageIndex{2}\): Determining the Electronic Structure of Anions
Selenium and iodine are two essential trace elements that form anions. Write the electron configurations of the anions.
Solution
Se 2– : [Ar]3 d 10 4 s 2 4 p 6
I – : [Kr]4 d 10 5 s 2 5 p 6
Exercise \(\PageIndex{2}\)
Write the electron configurations of a phosphorus atom and its negative ion. Give the charge on the anion.
- Answer
-
P: [Ne]3 s 2 3 p 3
P 3– : [Ne]3 s 2 3 p 6
In ionic compounds, electrons are transferred between atoms of different elements to form ions. But this is not the only way that compounds can be formed. Atoms can also make chemical bonds by sharing electrons between each other. Such bonds are called covalent bonds . Covalent bonds are formed between two atoms when both have similar tendencies to attract electrons to themselves (i.e., when both atoms have identical or fairly similar ionization energies and electron affinities). For example, two hydrogen atoms bond covalently to form an H 2 molecule; each hydrogen atom in the H 2 molecule has two electrons stabilizing it, giving each atom the same number of valence electrons as the noble gas He.
Compounds that contain covalent bonds exhibit different physical properties than ionic compounds. Because the attraction between molecules, which are electrically neutral, is weaker than that between electrically charged ions, covalent compounds generally have much lower melting and boiling points than ionic compounds. In fact, many covalent compounds are liquids or gases at room temperature, and, in their solid states, they are typically much softer than ionic solids. Furthermore, whereas ionic compounds are good conductors of electricity when dissolved in water, most covalent compounds, being electrically neutral, are poor conductors of electricity in any state.
Formation of Covalent Bonds
Video \(\PageIndex{4}\) : What are covalent bonds?
Nonmetal atoms frequently form covalent bonds with other nonmetal atoms. For example, the hydrogen molecule, H 2 , contains a covalent bond between its two hydrogen atoms. Figure \(\PageIndex{3}\) illustrates why this bond is formed. Starting on the far right, we have two separate hydrogen atoms with a particular potential energy, indicated by the red line. Along the x -axis is the distance between the two atoms. As the two atoms approach each other (moving left along the x -axis), their valence orbitals (1 s ) begin to overlap. The single electrons on each hydrogen atom then interact with both atomic nuclei, occupying the space around both atoms. The strong attraction of each shared electron to both nuclei stabilizes the system, and the potential energy decreases as the bond distance decreases. If the atoms continue to approach each other, the positive charges in the two nuclei begin to repel each other, and the potential energy increases. The bond length is determined by the distance at which the lowest potential energy is achieved.
Figure \(\PageIndex{3}\) : The potential energy of two separate hydrogen atoms (right) decreases as they approach each other, and the single electrons on each atom are shared to form a covalent bond. The bond length is the internuclear distance at which the lowest potential energy is achieved.
It is essential to remember that energy must be added to break chemical bonds (an endothermic process), whereas forming chemical bonds releases energy (an exothermic process). In the case of H 2 , the covalent bond is very strong; a large amount of energy, 436 kJ, must be added to break the bonds in one mole of hydrogen molecules and cause the atoms to separate:
\[\ce{H2}(g)⟶\ce{2H}(g)\hspace{20px}ΔH=\mathrm{436\:kJ}\]
Conversely, the same amount of energy is released when one mole of H 2 molecules forms from two moles of H atoms:
\[\ce{2H}(g)⟶\ce{H2}(g)\hspace{20px}ΔH=\mathrm{−436\:kJ}\]
Pure vs. Polar Covalent Bonds
If the atoms that form a covalent bond are identical, as in H 2 , Cl 2 , and other diatomic molecules, then the electrons in the bond must be shared equally. We refer to this as a pure covalent bond . Electrons shared in pure covalent bonds have an equal probability of being near each nucleus. In the case of Cl 2 , each atom starts off with seven valence electrons, and each Cl shares one electron with the other, forming one covalent bond:
\[\ce{Cl + Cl⟶Cl2}\]
The total number of electrons around each individual atom consists of six nonbonding electrons and two shared (i.e., bonding) electrons for eight total electrons, matching the number of valence electrons in the noble gas argon. Since the bonding atoms are identical, Cl 2 also features a pure covalent bond.
When the atoms linked by a covalent bond are different, the bonding electrons are shared, but no longer equally. Instead, the bonding electrons are more attracted to one atom than the other, giving rise to a shift of electron density toward that atom. This unequal distribution of electrons is known as a polar covalent bond , characterized by a partial positive charge on one atom and a partial negative charge on the other. The atom that attracts the electrons more strongly acquires the partial negative charge and vice versa. For example, the electrons in the H–Cl bond of a hydrogen chloride molecule spend more time near the chlorine atom than near the hydrogen atom. Thus, in an HCl molecule, the chlorine atom carries a partial negative charge and the hydrogen atom has a partial positive charge. Figure \(\PageIndex{4}\) shows the distribution of electrons in the H–Cl bond. Note that the shaded area around Cl is much larger than it is around H. Compare this to Figure \(\PageIndex{3}\), which shows the even distribution of electrons in the H 2 nonpolar bond.
Figure \(\PageIndex{4}\) : (a) The distribution of electron density in the HCl molecule is uneven. The electron density is greater around the chlorine nucleus. The small, black dots indicate the location of the hydrogen and chlorine nuclei in the molecule. (b) Symbols δ+ and δ– indicate the polarity of the H–Cl bond.
We sometimes designate the positive and negative atoms in a polar covalent bond using a lowercase Greek letter “delta,” δ, with a plus sign or minus sign to indicate whether the atom has a partial positive charge (δ+) or a partial negative charge (δ–). This symbolism is shown for the H–Cl molecule in Figure \(\PageIndex{4b}\).
Summary
Video \(\PageIndex{5}\) : A review of atomic bonding.
Atoms gain or lose electrons to form ions with particularly stable electron configurations. The charges of cations formed by the representative metals may be determined readily because, with few exceptions, the electronic structures of these ions have either a noble gas configuration or a completely filled electron shell. The charges of anions formed by the nonmetals may also be readily determined because these ions form when nonmetal atoms gain enough electrons to fill their valence shells.
Covalent bonds form when electrons are shared between atoms and are attracted by the nuclei of both atoms. In pure covalent bonds, the electrons are shared equally. In polar covalent bonds, the electrons are shared unequally, as one atom exerts a stronger force of attraction on the electrons than the other.
Glossary
- inert pair effect
- tendency of heavy atoms to form ions in which their valence s electrons are not lost
- ionic bond
- strong electrostatic force of attraction between cations and anions in an ionic compound
- bond length
- distance between the nuclei of two bonded atoms at which the lowest potential energy is achieved
- covalent bond
- bond formed when electrons are shared between atoms
Contributors and Attributions
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).
- Adelaide Clark, Oregon Institute of Technology
- Fuse School, Open Educational Resource free of charge, under a Creative Commons License: Attribution-NonCommercial CC BY-NC (View License Deed: https://creativecommons.org/licenses/by-nc/4.0/ )
- Crash Course Chemistry: Crash Course is a division of Complexly and videos are free to stream for educational purposes.
- Sci Show is a division of Complexly and videos are free to stream for educational purposes.
- TED-Ed’s commitment to creating lessons worth sharing is an extension of TED’s mission of spreading great ideas. Within TED-Ed’s growing library of TED-Ed animations, you will find carefully curated educational videos, many of which represent collaborations between talented educators and animators nominated through the TED-Ed website.


9.6B: How Skeletal Muscles Are Named
The anatomical arrangement of skeletal muscle fascicles can be described as parallel, convergent, pennate, or sphincter.
- Differentiate among parallel, pennate, convergent, and sphincter muscle types
Key Points
- Parallel muscles are the most abundant and typical, with fascicles arranged parallel to one another.
- Convergent muscles are similar to parallel muscles in attachment, although fascicles do not run parallel to one another, producing a broader muscle.
- In a pennate muscle the tendon runs through the length of the muscle, with the fascicles attaching at an angle.
- Sphincter muscles are characterized by a circular arrangement of fascicles around an opening. With contraction, the opening becomes smaller.
Key Terms
- Parallel : A muscle with a common point of attachment, with fascicles running parallel to each other.
- Circular : A ring like band of muscle that surrounds a bodily opening, constricting and relaxing to control flow.
- Pennate : A feather shaped muscle with fascicles that attach obliquely (at an angle) to a central tendon.
- Convergent : A muscle with a common point of attachment, although individual fascicles do not necessarily run parallel to each other.
Skeletal muscle can be categorised into four groups based on its anatomical arrangement.
Parallel
Parallel muscles are characterized by fascicles that run parallel to one another, and contraction of these muscle groups acts as an extension of the contraction of a single muscle fiber. Most skeletal muscles in the body are parallel muscles; although they can be seen in a variety of shapes such as flat bands, spindle shaped, and some can have large protrusions in their middle known as the belly of the muscle.
Parallel muscles can be divided into fusiform and non-fusiform types based on their shape. Fusiform muscles are more spindle shaped (their diameter at the center is greater than at either end), whereas, non-fusiform muscles are more rectangular with a constant diameter.
The biceps brachii is an example of a fusiform parallel muscle, and is responsible for flexing the forearm.
Convergent
Convergent muscles have a common point of attachment, from which the muscle fascicles extend outward, not necessarily in a specific spatial pattern, allowing the muscle to cover a broad surface. These muscles do not tend to exert as much force on their tendons. Muscle fibers can often exert opposing effects during contraction, such as not pulling in the same direction depending on the location of the muscle fiber. Covering a broad surface these fibers allow for more versatile types of movement. Because the fascicles pull on the tendons at an angle, they do not move the tendon as far as their parallel muscle counterparts. Despite this they generate greater tension because they possess a greater amount of muscle fibers than similarly sized parallel muscles.
The pectoralis major found in the chest is an example of a convergent muscle, and is responsible for flexing the upper arm.
Pennate
In Pennate muscles, the tendon runs through the length of the muscle. Fascicles pull on the tendon at an angle, thus not moving as far at the parallel muscles during a contraction. However, these muscles tend to have relatively more muscle fibers than similarly sized parallel muscles, and thus carry more tension.
If all the fascicles of a pennate muscle are on the same side of the tendon, the pennate muscle is called unipennate. If the fascicles lie to either side of the tendon the muscle is called bipennate. If the central tendon branches within a pennate muscle, the muscle is called multipennate.
The rectus femoris found in the thigh, and responsible for its flexion, is an example of a bipennate muscle.
Circular
The fibers of the circular or sphincter muscles are arranged concentrically around an opening or recess. As the muscle contracts, the opening it circumvents gets smaller. For this reason, these muscles are often found at the entrances and exits of external and internal passageways. Skeletal circular muscles are different from smooth muscle equivalents due to their structure and because they are under voluntary control
The orbicularis oris which controls the opening of the mouth is an example of a circular muscle.
Types of muscle in the body : The four types of muscle; parallel (fusiform and non-fusiform), circular, convergent and pennate (uni, bi and multi).


13.15: Fertilization
In most fish and amphibians (frogs and toads) fertilization of the egg cells takes place outside the body. The female lays the eggs and then the male deposits his sperm on or at least near them. In reptiles and birds, eggs are fertilized inside the body when the male deposits the sperm inside the egg duct of the female. The egg is then surrounded by a resistant shell, “laid” by the female and the embryo completes its development inside the egg.
In mammals the sperm are placed in the body of the female and the eggs are fertilized internally. They then develop to quite an advanced stage inside the body of the female. When they are born they are fed on milk excreted from the mammary glands and protected by their parents until they become independent.


4.4: Arithmetic-Geometric Inequality
We have presented modest generalizations of basic mathematical induction (Corollary \(4.8\) and Corollary 4.9). The formality of our approach might suggest that induction is a rigid technique that must be applied inflexibly in a specific prescriptive way. To a mathematician induction is governed by two ideas:
(1) Induction uses the well ordering of the natural numbers, or more generally any well-ordered set, to prove universal statements quantified over the set.
(2) Every element in the set over which you quantify must be accounted for by the induction.
The formal characterizations of induction in Section \(4.2\) are sufficient but not necessary to achieve the objectives of a proof by induction. The theorem in this section will give you a sense about how the technique of induction can be extended. DEFINITION. Arithmetic mean Let \(a_{1}, \ldots, a_{N}\) be real numbers. The arithmetic mean of \(a_{1}, \ldots, a_{N}\) is \[\frac{1}{N}\left(\sum_{n=1}^{N} a_{n}\right) .\] Definition. Geometric mean Let \(a_{1}, \ldots, a_{N}\) be positive real numbers. The geometric mean of \(a_{1}, \ldots, a_{N}\) is \[\sqrt[N]{a_{1} \cdots a_{N}} .\] THEOREM 4.16. Arithmetic-geometric mean inequality Let \(a_{1}, \ldots, a_{N} \in\) \(\mathbb{R}^{+}\) . Then \[\sqrt[N]{a_{1} \cdots a_{n}} \leq \frac{1}{N}\left(\sum_{n=1}^{N} a_{n}\right) .\] DisCUSSION. We prove this with an interesting argument due originally to Cauchy; our treatment is from the book [1]. We argue by induction on the size of the sample over which we are computing the means. After arguing the base case we show that if the inequality holds for the arithmetic and geometric means of \(N\) numbers, it necessarily holds for the means of \(2 N\) numbers. This implies that the theorem holds for the means of \(2^{N}\) numbers for any \(N \in \mathbb{N}\) (by a standard induction argument).
We then show that the result holding for \(N\) numbers implies that it holds for \(N-1\) numbers. This implies that if the result holds at a natural number \(N\) , the inequality holds for all means of fewer than \(N\) numbers. Given any \(k \in \mathbb{N}, 2^{k}>k\) and since the theorem holds for means of \(2^{k}\) numbers, it holds for means of \(k\) terms.
Proof. We argue by induction on the number of terms on each side of the inequality.
Base case: \((N=2)\)
Let \(a_{1}, a_{2} \in \mathbb{R}^{+}\) . Then \[\left(a_{1}-a_{2}\right)^{2}=a_{1}^{2}-2 a_{1} a_{2}+a_{2}^{2} \geq 0 .\] Therefore \[2 a_{1} a_{2} \leq a_{1}^{2}+a_{2}^{2},\] and \[\begin{aligned} 4 a_{1} a_{2} & \leq a_{1}^{2}+2 a_{1} a_{2}+a_{2}^{2} \\ &=\left(a_{1}+a_{2}\right)^{2} . \end{aligned}\] Thus \[2 \sqrt{a_{1} a_{2}} \leq a_{1}+a_{2} .\] Therefore the inequality holds for two terms.
Induction step:
Let \(P(N)\) be the statement that (4.17) holds for all \(a_{1}, \ldots, a_{N}>0\) . We show that \(P(N) \Rightarrow P(2 N)\) . Let \[G_{N}=\prod_{n=1}^{N} a_{n}\] and \[A_{N}=\left(\frac{\sum_{n=1}^{N} a_{n}}{N}\right) .\] So \[\begin{aligned} G_{2 N} &=\prod_{n=1}^{2 N} a_{n} \\ &=\left(\prod_{n=1}^{N} a_{n}\right)\left(\prod_{n=N+1}^{2 N} a_{n}\right) \\ & \leq_{I H}\left(\sum_{n=1}^{N} \frac{a_{n}}{N}\right)^{N}\left(\sum_{n=N+1}^{2 N} \frac{a_{n}}{N}\right)^{N} \end{aligned}\] Let \[B=\sum_{n=N+1}^{2 N} \frac{a_{n}}{N} .\] By the base case \[\begin{aligned} A_{N} B & \leq\left(\frac{A_{N}+B}{2}\right)^{2} \\ &=\left(A_{2 N}\right)^{2} \end{aligned}\] So \[\begin{aligned} \left(A_{N}\right)^{N} B^{N} &=\left(A_{N} B\right)^{N} \\ & \leq\left(\left(A_{2 N}\right)^{2}\right)^{N} \\ &=\left(A_{2 N}\right)^{2 N} \end{aligned}\] Thus \[G_{2 N} \leq\left(A_{2 N}\right)^{2 N} .\] Therefore, for any \(N \in \mathbb{N}^{+}\) , \[P(N) \Rightarrow P(2 N) .\] Discussion. Let \(Q(N)\) be the statement \(P\left(2^{N}\right)\) . Then the argument thus far is a standard proof by induction of \(\left(\forall N \in \mathbb{N}^{+}\right) Q(N)\) . Of course we wish to show \((\forall N \in \mathbb{N}) P(N)\) . We do this by proving \[\left(\forall N \in \mathbb{N}^{+}\right) P(N+1) \Rightarrow P(N) .\] Let \(N>2\) . We prove that \[P(N+1) \Rightarrow P(N) .\] Assume \(P(N+1)\) . Then \[\left(G_{N}\right)\left(A_{N}\right) \leq\left(\frac{\left(\sum_{n=1}^{N} a_{n}\right)+A_{N}}{N+1}\right)^{N+1} .\] DiscuSsion. Recall that \(G_{N}\) is the product of \(a_{1}, \ldots, a_{N}\) . We are treating the sum \(A_{N}\) as the \(N+1^{\text {st }}\) factor, \(a_{N+1}\) , and applying the inequality \(P(N+1)\) .
As \[\begin{aligned} \left(\frac{\left(\sum_{n=1}^{N} a_{n}\right)+A_{N}}{N+1}\right)^{N+1} &=\left(\frac{N A_{N}+A_{N}}{N+1}\right)^{N+1} \\ &=\left(A_{N}\right)^{N+1} \end{aligned}\] Inequality \(4.18\) gives \[G_{N} A_{N} \leq A_{N}^{N+1}\] and so \[G_{N} \leq\left(A_{N}\right)^{N}\] which is the statement \(P(N)\) . So \[\left(\forall N \in \mathbb{N}^{+}\right) P(N+1) \Rightarrow P(N) .\] Hence for all \(N \geq 2, P(N)\) .
The arithmetic mean and geometric mean are different ways of understanding averages. They are related by the arithmetic geometric mean inequality (called the AGM inequality). Can we apply the inequality? Let’s consider an easy geometrical application of the case \(N=2\) . Consider the rectangle with sides length \(a\) and \(b\) . The perimeter of the rectangle is \[P=2 a+2 b\] and the area is \[A=a b .\]
In calculus you proved that the rectangle of fixed perimeter with the greatest area is the square. This can also be proved directly from the AGM inequality: \[\begin{aligned} P &=2 a+2 b \\ &=\frac{4 a+4 b}{2} \\ & \geq \sqrt{16 a b} \\ &=4 \sqrt{a b} . \end{aligned}\] So \[\frac{P^{2}}{16} \geq a b=A .\] Recall that \(P\) is fixed, and therefore so is \(\frac{P^{2}}{16}\) , and we have shown that this is an upper bound for the area of the rectangle.
Is this upper bound achieved? The area \(A\) of the rectangle varies according to the dimensions of the rectangle and if \(a=b\) \[\frac{P^{2}}{16}=\frac{(4 a)^{2}}{16}=A .\] Thus the maximum area of the rectangle is achieved when \(a=b\) . This result can be generalized to higher dimensions - without the need for multivariable calculus.
Proving theorems is not just a question of technique, though this must be mastered. It also requires creativity and insight. A beautiful collection of proofs is contained in the book [1] by Martin Aigner and Günter Ziegler.


12.10: Primary Key
Learning Objectives
- Discuss the importance of primary key uniqueness
- Choose a primary key
Every table must have a primary key—a field with a unique value for each record. Otherwise there would be no way to distinguish between records. A primary key consists of one or more fields that uniquely identify each record that you store in the table. Often, there is a unique identification number, such as an ID number, a serial number, or a code, that serves as a primary key.
For example, you might have a Customers table where each customer has a unique customer ID number. The customer ID field is the primary key of the Customers table. When a primary key contains more than one field, it is usually composed of pre-existing fields that, taken together, provide unique values. For example, you might use a combination of last name, first name, and birth date as the primary key for a table about people.
Choosing a Primary Key
Without the ProductID field, another field would be needed to uniquely identify each product in the database. Without a strong primary key, the database engine would be at a loss to perform basic operations on the table properly.
Good primary keys must:
- Uniquely identify each record,
- Not be null,
- Exist when the record is created,
- Must remain stable,
- Be simple and contain as few attributes as possible.
Contributors and Attributions
- Primary Key. Authored by : Robert Danielson. Provided by : Lumen Learning. License : CC BY: Attribution


5.4: To Be (an) Extra
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- Vicki Mayer
- University of California Press
“It’s the trend to work on the set, to be on the set, to be an extra, or have a friend that was an extra,” said an African American student who grew up in an affluent uptown neighborhood. The excitement, he posited, came from the narrative itself.
“The show has the potential to be truthful and realistic to the city,” he said. This appeal to the real and to fundamental truth-telling about the city in the post-K era was a frequent logic for joining the production. In this, extras were no different from fans in loving the show’s careful attention to vernacular culture, regional musical and culinary traditions, and painstakingly accurate archive of the urban cultural geography. Moreover, extras described, in very emotional ways, the solidarity they felt with the series creators in telling the story of a city under threat of disintegration. As a recent resident, a retired professor who became an extra, explained, he felt like he became an insider to the city’s trauma by watching Treme weekly in New Orleans with a group of Katrina survivors:
That experience certainly changed [me and my wife’s] relationship to the show both in terms of the knowledge gained but also a sympathy towards it. People talked about how, you know, in the opening credits, there’s the patterns of mold, and people said, “Yeah, that one looks like the one I have in my [flooded] house.” And so you get connected to the show in ways that are very unusual. But Treme has been and continues to be this booster for New Orleans as a city. And right after Katrina that was critical. So I was a worshipper of Treme at that time because I felt people had given up on New Orleans, I mean really had given up.
Here the speaker’s desire to work for the show was imbued with a near spiritual investment in the city as portrayed through the program. Inevitably, the labor of being an extra could never live up to these lofty aspirations of being real, telling the truth, and satisfying souls.
The emotional solidarity with Treme served the labor needs of the production well. They stemmed from an imagined belonging, first, to a community of empathy with the residents of a traumatized city and, second, to a television program imbued with the agency to help in the recovery. By generating a community of empathy with New Orleans, the number of people who could imagine themselves doing extra work stretched beyond a period of residency or even the geography of city boundaries, bringing in new residents and even regular pilgrims to the city. These people could believe in New Orleans as a standpoint or a way of being expressed in a popular T-shirt: “Be A New Orleanian Wherever You Are.” The motto, which recognizes the real exile of New Orleanians throughout the world, could also recognize anyone who felt the same passion as Simon and Overmyer did. One of my interviewees, commenting on a nonnative friend who regularly appeared on the program, called these extras the “super–New Orleanians,” those residents who “go to everything more than the people born here. They are the ones who know the musicians. They have all the connections. They are kind of in love with something they want to embrace much more than in the natural way. . . . They can be almost arrogant about the real New Orleans.” These extras put to work their shared dispositions toward the city precisely by being the kinds of culture bearers and bards that the show spotlighted as authentic. The program succeeded in promoting and preserving this vision of the city and its citizenry in part because so many people were willing to be part of the drama’s background.
For extras, however, their personal experiences were in the foreground when they considered why they would take off work or spend their leisure time working at the bottom of the organizational hierarchy on a Hollywood production. People spoke of how Treme had become part of their own traumatic stories: “I guess just living here and having been through Katrina and coming back. The first two months were so intense, and I think the show shows that in a really accurate way, just how intense everything was” (female, thirty-seven, housekeeper). “The show was close to home. There was a connection” (female, thirty, tour guide). In these quotes, the locational shooting of Treme became part of their own temporal and geographic locations: respectively, Katrina and home. This close siting of the series meant for some people that they felt they had to appear on the program. Extras spoke of a kind of doppelgänger effect, in which they already saw themselves in the story, and thus wanted to memorialize it. A middle-aged man who lived through the storm spoke of audibly laughing or sobbing through scenes, replaying them as he did his own memories: “I kept expecting to see myself in the background because the scenes were so real to me. . . . I think there’s some weird thing in my brain that I think I’m already a part of it. I think that would be really neat to be historically there and on film, to be part of New Orleans” (male, forty-seven, barista and composer).
This emotional and psychological investment in the series spoke to a heightened expectation for Treme that locals did not express for other productions around the city, even those specifically focused on Katrina. On one hand, the investment spoke to the role of media in memorializing tragedies. Amanda Lagerkvist, for example, has written about how television recurrently reminds viewers of its own heroic role not only in broadcasting the tragic events of September 11th to global audiences but also in dealing with the traumatic aftermath in nationally specific ways. 9 Seeing their lives unfold on the screen, Treme viewers wanted to be in the program, as if to merge the lived and its representation. One interviewee, a local musician, related her uncanny feelings about the program and her desire to work for it to looking at a Beatles album: “You’re just looking around. It’s like when I saw the big protest march in the last episode, I just keep seeing all these people from different parts of my life. They were all there. They were all extras so I joked that it was kinda like looking at a Sergeant Pepper album cover, you know, to see all these people you recognize” (female, forty-seven, nonprofit worker and singer). Extras wanted to not only “be in it because everyone else is in it,” as one said, but to be remembered as having been in it with everyone . Extras merged the politics of their labor recognition with the politics of belonging to the city and its historical record.
Treme ’s labor market and strategy for so many extras thus relied on larger processes of popular memorialization, which on this particular program, championed the “super–New Orleanian” as the authentic representative of the city’s recent past. As proxies for the city’s local culture, extras were tasked with being in and engaging with the city in a way that others would see as authentic. According to one tour guide who did extra work during a slow period, this aspect of the job was hardly a burden: “One of the days I did extra work I was down on Frenchmen Street, which I go to all the time, and I went to the Spotted Cat [music club] and watched the Jazz Vipers. . . . Now [in season two] a lot of my buddies have been on the show, so chances are if I do it again, I’m going to hang out with them and get paid for it.” The proposition of getting paid to hang out takes a postmodern spin on the idea of labor, as if being an extra is not really working or somehow subverting real work. At the same time, it was where and with whom that imbued the extras with an exchange value in the first instance. Producers did not need the extras to do anything but hang around with others who could give credence to the authenticity claims for New Orleans as a particular kind of place, where people congregate every day in dark, musty alcoves animated by old-timey jazz riffs and refrains.
Beyond work on the set, the personal commitment on the part of avid viewers transformed extra work into a political project to do more hours and types of media labor off the set. Although production crews frequently refer to themselves as a “family,” in the case of Treme , the city at large was often seen as part of an extended production family. Crew members volunteered themselves and solicited others to manage charity and thank-you events in some of the neighborhoods with heavy location shooting. In an era of compulsory volunteerism at work, 10 many extras saw free labor as a way to build their social network in the film industry, while being recognized by others as a participant in Treme ’s moral economy. Collecting Facebook likes and cheers of recognition at the bar screening consolidated the public meaningfulness of appearing on the show with a veneer of participating in the preservation of local culture. By equating being on set to hear some music with urban recovery, they could pretend that watching the show, being on the show, and then promoting the show through social networks would sustain other local circuits of cultural production.
Meanwhile, for some extras, familial tensions emerged. Not everyone could embody the kinds of authenticity sought by the program. A white university student hailing from the East Coast claimed she was picked immediately and placed prominently in the camera’s view, while many people of color with less capital were turned away. Conversely, extras reported that crews excluded white extras when producers decided that the location should be African American. These occurrences led to various conversations about the politics of race in the production versus the city. Although no one would argue that there are still many segregated spaces in the city, production decisions at times clashed with extras’ expectations of realism and truth-telling. Some of these conversations expressed bemusement, as if to reaffirm the extras’ local knowledge contra the creators. In one instance, two white, middle-class retirees who had spent the day waiting with a crowd of hundreds for a restaging of a music festival pondered why the staff would have brought in buses of black, working-class school kids. Citing the increasingly unaffordable ticket prices and the overwhelmingly white audiences at the festival since Katrina, the interlocutors wondered privately if the decision was aspirational, a vision of how the festival should be populated. Other decisions about extras were more poignant, leading to hurt feelings when producers told extras that they did not belong in a restaging of a moment from their own lives.
If extras had a sense of the uncanny watching the show, they also had moments when their memories were edited for the screen. An extended example illustrates the disconnect on a production that attracted workers with a sense of connectedness. I met one of these super–New Orleanians at a public discussion on Treme that I led at a coffeehouse. She was a white woman in her fifties. I knew her from several moments in my own cultural repertoire in the city, and I knew that she knew much of the informal cultural economy portrayed in Treme . Her regular presence in the Treme neighborhood was also felt in public Treme screenings, which in the first season allowed non-HBO subscribers to see the program. Treme cast members and production personnel were known to drop in. After one powerful screening, the crew combed the crowd to enlist extras for an upcoming scene drawn from real events. The woman recalled telling the crew she was there the night Glen David Andrews was arrested. “[The producers] didn’t really believe it too much, you know. They thought that was kinda strange, but [one of the mothers of the band members] came out and said she was glad I was there. That meant more to me than the money.” Here the slight inflicted by the producers was recouped by another member of the crowd, a Treme viewer who was also a Treme resident. The incident illustrates the difficulties for extras who gave of themselves only to find that Treme did not reciprocate or took too much in return.


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Central Limit Theorem
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Given a random variable with known mean μ and known standard deviation, σ, we are sampling with size n, and we are interested in two new RVs: the sample mean, \(\overline X\). If the size (\(n\)) of the sample is sufficiently large, then \(\overline{X} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)\). If the size (\(n\)) of the sample is sufficiently large, then the distribution of the sample means will approximate a normal distributions regardless of the shape of the population. The mean of the sample means will equal the population mean. The standard deviation of the distribution of the sample means, \(\frac{\sigma}{\sqrt{n}}\), is called the standard error of the mean.
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Mean
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A number that measures the central tendency; a common name for mean is "average." The term "mean" is a shortened form of "arithmetic mean." By definition, the mean for a sample (denoted by \(\overline x\)) is \(\overline x =\overline{x}=\frac{\text { Sum of all values in the sample }}{\text { Number of values in the sample }}\), and the mean for a population (denoted by \(\mu\)) is \(\mu=\frac{\text { Sum of all values in the population }}{\text { Number of values in the population }}\).
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Normal Distribution
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Notation: \(X \sim N(\mu, \sigma)\) for a continuous random variable, where \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation. If \(\mu = 0\) and \(\sigma = 1\), the random variable, \(z\), is called the
standard normal distribution
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Sampling Distribution
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Given simple random samples of size \(n\) from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution.
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Standard Error of the Mean
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The standard deviation of the distribution of the sample means, or \(\frac{\sigma}{\sqrt{n}}\)


2.8: Economic Stages
What you’ll learn to do: identify and explain the four stages of an economy (expansion, peak, contraction, and trough), and describe their impact on business operations
Spring, summer, fall and winter: the four seasons of the year. Expansion, peak, contraction, trough: the four seasons of an economy. In this next section you’ll learn about the cyclical nature of economies and how each of these “seasons” affects business operations.
Learning Objectives
- Differentiate between expansion, recession, and depression
Stages of the Economy
Economic Cycle
The term economic cycle (or boom-bust cycle) refers to economy-wide fluctuations in production, trade, and general economic activity. From a conceptual perspective, the economic cycle is the upward and downward movements of levels of GDP (gross domestic product) and refers to periods of expansion and contraction in the level of economic activities (business fluctuations) around a long-term growth trend.
Stages of the Economy
Economic cycles are identified as having four distinct economic stages: expansion, peak, contraction, and trough.
An expansion is characterized by increasing employment, economic growth, and upward pressure on prices. A peak is the highest point of the business cycle, when the economy is producing at maximum allowable output, employment is at or above full employment, and inflationary pressures on prices are evident. Following a peak, the economy typically enters into a correction which is characterized by a contraction where growth slows, employment declines (unemployment increases), and pricing pressures subside. The slowing ceases at the trough and at this point the economy has hit a bottom from which the next stage of expansion and contraction will emerge. In the United States, it is generally accepted that the National Bureau of Economic Research (NBER) is the final arbiter of the dates of the peaks and troughs of the economic cycle.
Since the economy is made up of businesses (both private and public), businesses are impacted by the stages of the economy or perhaps they cause the stages of the economy – or maybe a little of both! When we move from talking about stages of the economy, the terms used to describe the business cycle differ slightly, but you will see that they are almost mirror images of the economic stages.
Business Cycle Fluctuations
Business cycle fluctuations occur around a long-term growth trend just like economic cycles, but unlike economic cycles they are measured in terms of the growth rate of real gross domestic product (Real GDP) . This does not mean that the GDP is imaginary, but rather that GDP does not take into account inflation. Instead, real gross domestic product is the inflation adjusted value of the goods and services produced by labor and property located in the United States.
An expansion is the period from a trough to a peak, and a recession is the period from a peak to a trough. The NBER identifies a recession as “a significant decline in economic activity spread across the economy, lasting more than a few months, normally visible in real GDP, real income, employment, industrial production.” This is significantly different from the commonly cited definition of a recession being signaled by two consecutive quarters of decline in real GDP. If the economy does not begin to expand again, then the economy may be considered to be in a state of depression .
Impact of the Economic Cycle on Business Operations
How the economic cycle affects business operations may be best explained by looking at how one business responds to these cycles. Normal Maintenance is a small business that provides a variety of construction services to homeowners. They specialize in roofing, deck installations, siding, and general home maintenance. They employ three full-time workers, who typically work forty hours per week for an average of twelve dollars per hour. The company has been in business in the same town for than twenty years and has a solid reputation for quality work and reliability.
Expansion
Normal Maintenance is busy and has recently had to turn down jobs because it lacks the capacity to do all the work offered. Homeowners now want to make home repairs and improvements which they had had to put off during the sour economy. With the economy improving, others are fixing up their homes to sell. Faced with so much demand, the owner of Normal Maintenance must decide whether to pay his existing workers overtime (which will increase the costs for each job and reduce profits) or hire additional workers. The competition for qualified construction labor is steep, and he is concerned that he will have to pay more than his usual rate of twelve dollars per hour or possibly get workers who are not as qualified as his current crew. He is, however, able to charge higher prices for his work because homeowners are experiencing long waits and delays getting bids and jobs completed. The owner purchases a new truck and invests in additional tools in order to keep up with the demand for services. Customers are willing to pay more than usual so they can get the work done. Business is expanding to such an extent that Normal Maintenance and its suppliers are starting to have trouble obtaining materials such as shingles and siding because the manufacturers have not kept pace with the economic expansion. In general, business is great for Normal Maintenance, but the expansion brings challenges.
Peak
At the peak of the business cycle, the economy can be said to be “overheated.” Despite hiring additional workers, the owner and crews of Normal Maintenance are working seven days a week and are still unable to keep up with demand. They can’t work any harder or faster. As a result, the crews are exhausted and the quality of their work is beginning to decline. Customers leave messages requesting work and services, but the owner is so busy he doesn’t return phone calls. Jobs are getting started and completed late as the crews struggle to cover multiple job sites. As a result, customer complaints are on the rise, and the owner is worried about the long-term reputation of the business. Neither the business nor the economy can sustain this level of activity, and despite the fact that Normal Maintenance is making great money, everyone is ready for things to let up a little.
Contraction
As the economy begins to contract, business begins to slow down for Normal Maintenance. They find that they are caught up on work and they aren’t getting so many phone calls. The owner is able to reduce his labor costs by cutting back on overtime and eliminate working on the weekends. When the phone does ring, homeowners are asking for bids on work—not just placing work orders. Normal Maintenance loses out on several jobs because their bids are too high. The company begins to look for new suppliers who can provide them with materials at a cheaper price so they can be more competitive. The building material companies start offering “deals” and specials to contractors in order to generate sales. In general, competition for work has increased and some of the businesses that popped up during the expansion are no longer in the market. In the short term the owner is confident that he has enough work to keep his crew busy, but he’s concerned that if things don’t pick up, he might have to lay off some of the less experienced workers.
Trough
On Monday morning, the crew of Normal Maintenance show up to work and the owner has to send them home: there’s no work for them. During the week before, they worked only three days, and the owner is down to his original crew of three employees. Several months ago he laid off the workers hired during the expansion. Although that was a difficult decision, the owner knows from hard experience that sometimes businesses fail not because their owners make bad decisions, but because they run out of money during recessions when there isn’t enough customer demand to sustain them. Without enough working capital to keep the doors open, some are forced to close down.
Representatives from supply companies are stopping by the office hoping to get an order for even the smallest quantity of materials. The new truck and tools that the owner purchased during the boom now sit idle and represent additional debt and costs. The company’s remaining work comes from people who have decided to fix up their existing homes because the economy isn’t good enough for them to buy new ones. The owner increases his advertising budget, hoping to capture any business that might be had. He is optimistic that Normal Maintenance will weather this economic storm—they’ve done it before—but he’s worried about his employees paying their bills over the winter.
The owner of Normal Maintenance has been in business for a long time, so he’s had some experience with the economic cycle. Though each stage has its stressors, he has learned to plan for them. One thing he knows is that the economy will eventually begin to expand again and run through the cycle all over again.
practice question \(\PageIndex{1}\)
If the economy is currently experiencing high growth and inflation combined with low unemployment, it is likely to be at which phase of the business cycle?
- trough
- expansion
- recession or contraction
- Answer
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b . Expansions tend to be characterized by positive GDP growth and decreased unemployment and increasing inflation.