2.1: Introduction to Matrices
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Learning Objectives
In this section, you will learn to:
- Add and subtract matrices.
- Multiply a matrix by a scalar.
- Multiply two matrices.
A matrix is a 2 dimensional array of numbers arranged in rows and columns. Matrices provide a method of organizing, storing, and working with mathematical information. Matrices have an abundance of applications and use in the real world. Matrices provide a useful tool for working with models based on systems of linear equations. We’ll use matrices in sections 2.2, 2.3, and 2.4 to solve systems of linear equations with several variables in this chapter.
Matrices are used in encryption, which we will explore in section 2.5 and in economic modelling, explored in section 2.6. We use matrices again in chapter 4, in optimization problems such as maximizing profit or revenue, or minimizing cost. Matrices are used in business for scheduling, routing transportation and shipments, and managing inventory.
Just about any application that collects and manages data can apply matrices. Use of matrices has grown as the availability of data in many areas of life and business has increased. They are important tools for organizing data and solving problems in all fields of science, from physics and chemistry, to biology and genetics, to meteorology, and economics. In computer science, matrix mathematics lies behind animation of images in movies and video games.
Computer science analyzes diagrams of networks to understand how things are connected to each other, such as relationships between people on a social website, and relationships between results in line search and how people link from one website to another. The mathematics to work with network diagrams comprise the field of “graph theory”; it relies on matrices to organize the information in the graphs that diagram connections and associations in a network. For example, if you use Facebook or Linked-In, or other social media sites, these sites use network graphs and matrices to organize your relationships with other users.
Introduction to Matrices
A matrix is a rectangular array of numbers. Matrices are useful in organizing and manipulating large amounts of data. In order to get some idea of what matrices are all about, we will look at the following example.
Example \(\PageIndex{1}\)
Fine Furniture Company makes chairs and tables at its San Jose, Hayward, and Oakland factories. The total production, in hundreds, from the three factories for the years 2014 and 2015 is listed in the table below.
| 2014 | 2015 | |||
| CHAIRS | TABLES | CHAIRS | TABLES | |
| SAN JOSE | 30 | 18 | 36 | 20 |
| HAYWARD | 20 | 12 | 24 | 18 |
| OAKLAND | 16 | 10 | 20 | 12 |
- Represent the production for the years 2014 and 2015 as the matrices A and B.
- Find the difference in sales between the years 2014 and 2015.
- The company predicts that in the year 2020 the production at these factories will be double that of the year 2014. What will the production be for the year 2020?
Solution
a) The matrices are as follows:
\[A=\left[\begin{array}{ll}
30 & 18 \\
20 & 12 \\
16 & 10
\end{array}\right] \nonumber \]
\[\mathrm{B}=\left[\begin{array}{ll}
36 & 20 \\
24 & 18 \\
20 & 12
\end{array}\right] \nonumber \]
b) We are looking for the matrix \(B - A\). When two matrices have the same number of rows and columns, the matrices can be added or subtracted entry by entry. Therefore, we get
\[\mathrm{B}-\mathrm{A}=\left[\begin{array}{ll}
36-30 & 20-18 \\
24-20 & 18-12 \\
20-16 & 12-10
\end{array}\right]=\left[\begin{array}{ll}
6 & 2 \\
4 & 6 \\
4 & 2
\end{array}\right] \nonumber \]
c) We would like a matrix that is twice the matrix of 2014, i.e., \(2A\).
Whenever a matrix is multiplied by a number, each entry is multiplied by the number.
\[2 \mathrm{A}=2\left[\begin{array}{ll}
30 & 18 \\
20 & 12 \\
16 & 10
\end{array}\right]=\left[\begin{array}{ll}
60 & 36 \\
40 & 24 \\
32 & 20
\end{array}\right] \nonumber \]
Before we go any further, we need to familiarize ourselves with some terms that are associated with matrices. The numbers in a matrix are called the entries or the elements of a matrix.
Whenever we talk about a matrix, we need to know the size or the dimension of the matrix. The dimension of a matrix is the number of rows and columns it has. When we say a matrix is a “3 by 4 matrix”, we are saying that it has 3 rows and 4 columns. The rows are always mentioned first and the columns second. This means that a \(3 \times 4\) matrix does not have the same dimension as a \(4 \times 3\) matrix.
\[A=\left[\begin{array}{cccc}
1 & 4 & -2 & 0 \\
3 & -1 & 7 & 9 \\
6 & 2 & 0 & 5
\end{array}\right] \nonumber \]
\[B=\left[\begin{array}{ccc}
2 & 9 & 8 \\
-3 & 0 & 1 \\
6 & 5 & -2 \\
-4 & 7 & 8
\end{array}\right] \nonumber \]
Matrix \(A\) has dimensions \(3 \times 4\) and matrix \(B\) has dimensions \(4 \times 3\).
A matrix that has the same number of rows as columns is called a square matrix . A matrix with all entries zero is called a zero matrix. A square matrix with 1's along the main diagonal and zeros everywhere else, is called an identity matrix . When a square matrix is multiplied by an identity matrix of same size, the matrix remains the same.
\[I=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right] \nonumber \]
Matrix \(I\) is a \(3 \times 3\) identity matrix
A matrix with only one row is called a row matrix or a row vector, and a matrix with only one column is called a column matrix or a column vector . Two matrices are equal if they have the same size and the corresponding entries are equal.
We can perform arithmetic operations with matrices. Next we will define and give examples illustrating the operations of matrix addition and subtraction, scalar multiplication, and matrix multiplication. Note that matrix multiplication is quite different from what you would intuitively expect, so pay careful attention to the explanation. Note also that the ability to perform matrix operations depends on the matrices involved being compatible in size, or dimensions, for that operation. The definition of compatible dimensions is different for different operations, so note the requirements carefully for each.
Matrix Addition and Subtraction
If two matrices have the same size, they can be added or subtracted. The operations are performed on corresponding entries.
Example \(\PageIndex{2}\)
Given the matrices \(A\), \(B\), \(C\) and \(D\), below
\[A=\left[\begin{array}{lll}
1 & 2 & 4 \\
2 & 3 & 1 \\
5 & 0 & 3
\end{array}\right] \quad B=\left[\begin{array}{lll}
2 & -1 & 3 \\
2 & 4 & 2 \\
3 & 6 & 1
\end{array}\right] \quad C=\left[\begin{array}{l}
4 \\
2 \\
3
\end{array}\right] \quad D=\left[\begin{array}{r}
-2 \\
-3 \\
4
\end{array}\right] \nonumber \]
Find, if possible.
- \(A + B\)
- \(C - D\)
- \(A + D.\)
Solution
As we mentioned earlier, matrix addition and subtraction involves performing these operations entry by entry.
a) We add each element of \(A\) to the corresponding entry of \(B\).
\[A+B=\left[\begin{array}{lll}
3 & 1 & 7 \\
4 & 7 & 3 \\
8 & 6 & 4
\end{array}\right] \nonumber \]
b) Just like the problem above, we perform the subtraction entry by entry.
\[\mathrm{C}-\mathrm{D}=\left[\begin{array}{c}
6 \\
5 \\
-1
\end{array}\right] \nonumber \]
c) The sum \(A + D\) cannot be found because the two matrices have different sizes.
Note: Two matrices can only be added or subtracted if they have the same dimension.
Multiplying a Matrix by a Scalar
If a matrix is multiplied by a scalar, each entry is multiplied by that scalar. We can consider scalar multiplication as multiplying a number and a matrix to obtain a new matrix as the product.
Example \(\PageIndex{3}\)
Given the matrix \(A\) and \(C\) in the example above, find \(2A\) and \(- 3C\).
Solution
To find \(2A\), we multiply each entry of matrix \(A\) by 2, and to find \(-3C\), we multiply each entry of C by -3. The results are given below.
a) We multiply each entry of A by 2.
\[2 \mathrm{A}=\left[\begin{array}{ccc}
2 & 4 & 8 \\
4 & 6 & 2 \\
10 & 0 & 6
\end{array}\right] \nonumber \]
b) We multiply each entry of C by -3.
\[-3 C=\left[\begin{array}{c}
-12 \\
-6 \\
-9
\end{array}\right] \nonumber \]
Multiplication of Two Matrices
To multiply a matrix by another is not as easy as the addition, subtraction, or scalar multiplication of matrices. Because of its wide use in application problems, it is important that we learn it well. Therefore, we will try to learn the process in a step by step manner. We first begin by finding a product of a row matrix and a column matrix.
Example \(\PageIndex{4}\)
Find the product \(AB\), given
\[A=\left[\begin{array}{lll}
2 & 3 & 4
\end{array}\right] \nonumber \]
and
\[B=\left[\begin{array}{l}
a \\
b \\
c
\end{array}\right]. \nonumber \]
Solution
The product is a \(1 \times 1\) matrix whose entry is obtained by multiplying the corresponding entries and then forming the sum.
\[\begin{align*}
\mathrm{AB} &=\left[\begin{array}{lll}
2 & 3 & 4
\end{array}\right]\left[\begin{array}{l}
\mathrm{a} \\
\mathrm{b} \\
\mathrm{c}
\end{array}\right] \\[4pt] &=[2 (\mathrm{a}+3 \mathrm{b}+4 \mathrm{c})]
\end{align*} \nonumber \]
Note that \(AB\) is a \(1 \times 1\) matrix, and its only entry is \(2a + 3b + 4c\).
Example \(\PageIndex{5}\)
Find the product \(AB\), given
\[A=\left[\begin{array}{lll}
2 & 3 & 4
\end{array}\right] \nonumber \]
and
\[B=\left[\begin{array}{l}
5 \\
6 \\
7
\end{array}\right] \nonumber \]
Solution
Again, we multiply the corresponding entries and add.
\[\begin{align*} \mathrm{AB} &=\left[\begin{array}{lll}
2 & 3 & 4
\end{array}\right]\left[\begin{array}{l}
5 \\
6 \\
7
\end{array}\right] \\[4pt] &=[2 \cdot 5+3 \cdot 6+4 \cdot 7]\\[4pt] &=[10+18+28]\\[4pt] &=[56] \end{align*} \nonumber \]
Note: In order for a product of a row matrix and a column matrix to exist, the number of entries in the row matrix must be the same as the number of entries in the column matrix.
Example \(\PageIndex{6}\)
Find the product AB, given
\[A=\left[\begin{array}{lll}
2 & 3 & 4
\end{array}\right] \nonumber \]
and
\[B=\left[\begin{array}{ll}
5 & 3 \\
6 & 4 \\
7 & 5
\end{array}\right]. \nonumber \]
Solution
We know how to multiply a row matrix by a column matrix. To find the product \(AB\), in this example, we will multiply the row matrix \(A\) to both the first and second columns of matrix \(B\), resulting in a \(1 \times 2\) matrix.
\[\mathrm{AB}=\left[\begin{array}{lll}
2 \cdot 5+3 \cdot 6+4 \cdot 7 & 2 \cdot 3+3 \cdot 4+4 \cdot 5
\end{array}\right]=\left[\begin{array}{ll}
56 & 38
\end{array}\right] \nonumber \]
We multiplied a \(1 \times 3\) matrix by a matrix whose size is \(3 \times 2\). So unlike addition and subtraction, it is possible to multiply two matrices with different dimensions, if the number of entries in the rows of the first matrix is the same as the number of entries in the columns of the second matrix.
Example \(\PageIndex{7}\)
Find the product \(AB\), given:
\[A=\left[\begin{array}{lll}
2 & 3 & 4 \\
1 & 2 & 3
\end{array}\right] \nonumber \]
and
\[\mathrm{B}=\left[\begin{array}{ll}
5 & 3 \\
6 & 4 \\
7 & 5
\end{array}\right] \nonumber \]
Solution
This time we are multiplying two rows of the matrix \(A\) with two columns of the matrix \(B\). Since the number of entries in each row of \(A\) is the same as the number of entries in each column of \(B\), the product is possible. We do exactly what we did in the last example. The only difference is that the matrix \(A\) has one more row.
We multiply the first row of the matrix \(A\) with the two columns of \(B\), one at a time, and then repeat the process with the second row of A. We get
\[\mathrm{AB}=\left[\begin{array}{lll}
2 & 3 & 4 \\
1 & 2 & 3
\end{array}\right]\left[\begin{array}{ll}
5 & 3 \\
6 & 4 \\
7 & 5
\end{array}\right]=\left[\begin{array}{ll}
2 \cdot 5+3 \cdot 6+4 \cdot 7 & 2 \cdot 3+3 \cdot 4+4 \cdot 5 \\
1 \cdot 5+2 \cdot 6+3 \cdot 7 & 1 \cdot 3+2 \cdot 4+3 \cdot 5
\end{array}\right] \nonumber \]
\[\mathrm{AB}=\left[\begin{array}{ll}
56 & 38 \\
38 & 26
\end{array}\right] \nonumber \]
Example \(\PageIndex{8}\)
Find, if possible:
- \(EF\)
- \(FE\)
- \(FH\)
- \(GH\)
- \(HG\)
\[\mathrm{E}=\left[\begin{array}{ll}
1 & 2 \\
4 & 2 \\
3 & 1
\end{array}\right] \quad \mathrm{F}=\left[\begin{array}{ll}
2 & -1 \\
3 & 2
\end{array}\right] \quad \mathrm{G}=\left[\begin{array}{lll}
4 & 1
\end{array}\right] \quad \mathrm{H}=\left[\begin{array}{l}
-3 \\
-1
\end{array}\right] \nonumber \]
Solution
a) To find \(EF\), we multiply the first row \(\left[\begin{array}{ll}
1 & 2
\end{array}\right]\)
of E with the columns \(\left[\begin{array}{l}
2 \\
3
\end{array}\right] \text { and }\left[\begin{array}{l}
1 \\
-2
\end{array}\right]\) of the matrix F, and then repeat the process by multiplying the other two rows of E with these columns of F. The result is as follows:
\[\mathrm{EF}=\left[\begin{array}{ll}
1 & 2 \\
4 & 2 \\
3 & 1
\end{array}\right]\left[\begin{array}{cc}
2 & -1 \\
3 & 2
\end{array}\right]=\left[\begin{array}{cc}
1 \cdot 2+2 \cdot 3 & 1 \cdot-1+2 \cdot 2 \\
4 \cdot 2+2 \cdot 3 & 4 \cdot-1+2 \cdot 2 \\
3 \cdot 2+1 \cdot 3 & 3 \cdot-1+1 \cdot 2
\end{array}\right]=\left[\begin{array}{cc}
8 & 3 \\
14 & 0 \\
9 & -1
\end{array}\right] \nonumber \]
b) Product \(FE\) is not possible because the matrix F has two entries in each row, while the matrix E has three entries in each column. In other words, the matrix F has two columns, while the matrix E has three rows.
c) \[\mathrm{FH}=\left[\begin{array}{cc}
2 & -1 \\
3 & 2
\end{array}\right]\left[\begin{array}{c}
-3 \\
-1
\end{array}\right]=\left[\begin{array}{c}
2 \cdot-3+-1 \cdot-1 \\
3 \cdot-3+2 \cdot-1
\end{array}\right]=\left[\begin{array}{c}
-5 \\
-11
\end{array}\right] \nonumber \]
d) \[\mathrm{GH}=\left[\begin{array}{ll}
4 & 1
\end{array}\right]\left[\begin{array}{l}
-3 \\
-1
\end{array}\right]=\left[\begin{array}{l}
4 \cdot-3+1 \cdot-1 \\
-1
\end{array}\right]=[-13] \nonumber \]
e) \[\mathrm{HG}=\left[\begin{array}{l}
-3 \\
-1
\end{array}\right]\left[\begin{array}{ll}
4 & 1
\end{array}\right]=\left[\begin{array}{ll}
-3 \cdot 4 & -3 \cdot 1 \\
-1 \cdot 4 & -1 \cdot 1
\end{array}\right]=\left[\begin{array}{cc}
-12 & -3 \\
-4 & -1
\end{array}\right] \nonumber \]
We summarize some important properties of matrix multiplication that we observed in the previous examples.
In order for product \(\bf{AB}\) to exist:
- the number of columns of \(\bf{A}\) must equal the number of rows of \(\bf{B}\)
- if matrix \(\bf{A}\) has dimension \(\bf{m \times n}\) and matrix \(\bf{B}\) has dimension \(\bf{n \times p}\) , then the product \(\bf{AB}\) will be a matrix with dimension \(\bf{m \times p}\) .
Matrix multiplication is not commutative: if both matrix products \(\bf{AB}\) and \(\bf{BA}\) exist, most of the time \(\bf{AB}\) will not equal \(\bf{BA}\) .
Example \(\PageIndex{9}\)
Given matrices \(R\), \(S\), and \(T\) below, find \(2RS - 3ST\).
\[R=\left[\begin{array}{lll}
1 & 0 & 2 \\
2 & 1 & 5 \\
2 & 3 & 1
\end{array}\right] \quad S=\left[\begin{array}{lll}
0 & -1 & 2 \\
3 & 1 & 0 \\
4 & 2 & 1
\end{array}\right] \quad T=\left[\begin{array}{lll}
-2 & 3 & 0 \\
-3 & 2 & 2 \\
-1 & 1 & 0
\end{array}\right] \nonumber \]
Solution
We multiply the matrices R and S.
\begin{aligned}
&\mathrm{RS}=\left[\begin{array}{ccc}
8 & 3 & 4 \\
23 & 9 & 9 \\
13 & 3 & 5
\end{array}\right]\\
&\begin{array}{l}
2 \mathrm{RS}=2\left[\begin{array}{ccc}
8 & 3 & 4 \\
23 & 9 & 9 \\
13 & 3 & 5
\end{array}\right]=\left[\begin{array}{ccc}
16 & 6 & 8 \\
46 & 18 & 18 \\
26 & 6 & 10
\end{array}\right] \\
\mathrm{ST}=\left[\begin{array}{ccc}
1 & 0 & -2 \\
-9 & 11 & 2 \\
-15 & 17 & 4
\end{array}\right] \\
3 \mathrm{ST}=3\left[\begin{array}{ccc}
1 & 0 & -2 \\
-9 & 11 & 2 \\
-15 & 17 & 4
\end{array}\right]=\left[\begin{array}{ccc}
3 & 0 & -6 \\
-27 & 33 & 6 \\
-45 & 51 & 12
\end{array}\right]
\end{array}
\end{aligned}
Thus
\[2 \mathrm{RS}-3 \mathrm{ST}=\left[\begin{array}{ccc}
16 & 6 & 8 \\
46 & 18 & 18 \\
26 & 6 & 10
\end{array}\right]-\left[\begin{array}{ccc}
3 & 0 & -6 \\
-27 & 33 & 6 \\
-45 & 51 & 12
\end{array}\right]=\left[\begin{array}{ccc}
13 & 6 & 14 \\
73 & -15 & 12 \\
71 & -45 & -2
\end{array}\right] \nonumber \]
Example \(\PageIndex{10}\)
Find \(F^2\) given matrix
\[\mathrm{F}=\left[\begin{array}{ll}
2 & -1 \\
3 & 2
\end{array}\right] \nonumber \]
Solution
\(F^2\) is found by multiplying matrix \(F\) by itself, using matrix multiplication.
\[\mathrm{F}^{2}=\left[\begin{array}{cc}
2 & -1 \\
3 & 2
\end{array}\right]\left[\begin{array}{cc}
2 & -1 \\
3 & 2
\end{array}\right]=\left[\begin{array}{cc}
2 \cdot 2+(-1) \cdot 3 & 2 \cdot(-1)+(-1) \cdot 2 \\
3 \cdot 2+2 \cdot 3 & 3 \cdot(-1)+2 \cdot 2
\end{array}\right]=\left[\begin{array}{cc}
1 & -4 \\
12 & 1
\end{array}\right] \nonumber \]
Note that \(F^2\) is not found by squaring each entry of matrix \(F\). The process of raising a matrix to a power, such as finding \(F^2\), is only possible if the matrix is a square matrix.
USING MATRICES TO REPRESENT A SYSTEM OF LINEAR EQUATIONS
In this chapter, we will be using matrices to solve linear systems. In section 2.4, we will be asked to express linear systems as the matrix equation \(\bf{AX = B}\), where \(A\), \(X\), and \(B\) are matrices.
- Matrix \(A\) is called the coefficient matrix.
- Matrix \(X\) is a matrix with 1 column that contains the variables.
- Matrix \(B\) is a matrix with 1 column that contains the constants.
Example \(\PageIndex{11}\)
Verify that the system of two linear equations with two unknowns:
\begin{array}{l}
a x+b y=h \\
c x+d y=k
\end{array}
can be written as \(AX = B\), where
\[A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \quad X=\left[\begin{array}{l}
x \\
y
\end{array}\right] \text { and } B=\left[\begin{array}{l}
h \\
k
\end{array}\right] \nonumber \]
Solution
If we multiply the matrices \(A\) and \(X\), we get
\[A X=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right]=\left[\begin{array}{c}
a x+b y \\
c x+d y
\end{array}\right] \nonumber \]
If \(AX = B\) then
\[\left[\begin{array}{l}
a x+b y \\
c x+d y
\end{array}\right]=\left[\begin{array}{l}
h \\
k
\end{array}\right] \nonumber \]
If two matrices are equal, then their corresponding entries are equal. It follows that
\begin{array}{l}
a x+b y=h \\
c x+d y=k
\end{array}
Example \(\PageIndex{12}\)
Express the following system as a matrix equation in the form \(AX = B\).
\begin{array}{l}
2 x+3 y-4 z=5 \\
3 x+4 y-5 z=6 \\
5 x \quad-6 z=7
\end{array}
Solution
This system of equations can be expressed in the form \(AX = B\) as shown below.
\[\left[\begin{array}{ccc}
2 & 3 & -4 \\
3 & 4 & -5 \\
5 & 0 & -6
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
5 \\
6 \\
7
\end{array}\right] \nonumber \]


3.3: Piaget's and Kohlberg's Cognitive Developmental Theory (Psychology)
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Piaget and Kohlberg studies the mental process children use to understand their observations and experiences. Children develop organizing categories called schemas. Sex is a very important schema for young children. Their interpretations of the world, of interactions, and of others are limited by their mental maturity. Early on children’s thinking tends to rely on simple (often visual) cues. Females and males look differently in Western culture. Think of how women and men often dress or are often represented in popular culture. What are some of the common characteristics of female/maleness on TV or in movies or children’s books? So children often rely on those “obvious” physical cues to differentiate between men and women.
Cognitive factors in children's understanding of gender and gender stereotypes may contribute to their acquisition of gender roles. Kohlberg's three-stage cognitive developmental theory of gender typing suggests that children begin by categorizing themselves as male or females with reinforcement from outsiders such parents, and then feel rewarded by behaving in gender-consistent ways from external means. 30 According to Kohlberg, children acquire gender roles after she/he has gained an understanding and awareness that her/his sex is permanent, constant, and will never change. Children who are highly gender schematic often have parents or caregivers, especially fathers, who give them a lot of positive and negative reinforcement when it comes to gender-related activities. This teaches children gender-type behaviors as encourages them to pay more attention to gender as a social organizing category. Gender constancy emerges somewhere between 3-7 years of age. 31
30
Martin, C. L., Ruble, D. N., & Szkrykablo, J. (2002). Cognitive theories of early gender development. American
psychological association. 4(23), 544–557
31
Ruble DN, Martin C. Gender development. In: Damon W, Eisenberg N, editors. Handbook of child psychology: Vol. 3,
Social, emotional, and personality development. Hoboken, NJ: Wiley; 1998. pp. 933–1016.


9.3.1: Genetic Disorders and the Ethics of Prenatal Testing
Genetic errors are relatively common in the process of conception, and are thought to be the cause of about 50% of known miscarriages (ACOG Committee on Practice Bulletins—Gynecology, 2021). Even if a fertilized egg implants successfully in the uterine wall and continues to grow, genetic mutations can cause a variety of problems. Since each embryo inherits half of their 46 chromosomes (DNA) from each parent, certain diseases can be inherited from only one parent if the genetic disorder is dominant - the parent will pass that trait on to approximately half of their offspring. Some examples include Huntington’s disease (a brain disease), and certain types of dwarfism (achondroplasia) and gigantism (Marfan syndrome). Recessive genes are associated with some diseases that will affect a child if both parents carry them, and if both of those recessive genes are passed on (Schneider, 2020). See Fig. \(\PageIndex{1}\) below for the “Punnett Square” diagram illustrating recessive genes.
Examples of diseases associated with recessive genes include:
- Tay-Sachs disease: a neurological disorder, most common in people with Ashkenazi Jewish ancestry;
- Sickle-cell anemia: causes red blood cells to be shaped like a crescent and decreases oxygen transportation, most common in people with African or African American ancestry;
- Cystic fibrosis: a disease of the lungs causing excessive mucus production and damage to lung tissue, most common in people with northern European ancestry;
- Thalassemia: which causes a lack of hemoglobin in the blood, decreasing oxygen transport capacity, affects those with Mediterranean, Middle Eastern, Asian, and African ancestry, (CDC, 2020, National Health Service, 2021, Schneider, 2020).
Still other genetic disorders are linked to the X chromosome or female sex chromosome. These diseases primarily affect genetic males who have XY chromosomes, since the affected X tends to be recessive (whereas in a genetic female with two X chromosomes, the recessive gene would not be expressed) (Schneider, 2020). For example, Duchenne muscular dystrophy is an X-linked disorder that leads to muscular degeneration, weakness, and eventually paralysis. It is not more common in any racial or ethnic populations, but much more common in genetic males than females (Muscular Dystrophy Association, 2017).
Some genetic disorders are not necessarily inherited but occur due to a genetic mishap in either the sperm, egg, or both prior to conception. The human gametes (egg and sperm) each contain one half of 23 pairs of chromosomes, and the resulting zygote (fertilized egg) will have 46 chromosomes, or 23 pairs. A trisomy occurs when there are 3 copies of one of the chromosomes, perhaps from the egg or the sperm carrying both chromosomes in a pair, instead of one. The most common expressions are trisomy 13 (Patau syndrome), trisomy 18 (Edwards syndrome) and trisomy 21 (Down syndrome). Infants born with trisomy 13 or 18 rarely survive past one year old, due to resulting malformations of the brain, heart, lungs, and kidneys (Stanford Medicine Children’s Health, n.d.). Down syndrome causes a spectrum of physical, mental, and developmental challenges, and can increase the risk of congenital heart defects, hearing loss, and sleep apnea. However, most people with Down syndrome can live long and healthy lives with the appropriate support (CDC, 2024d). A monosomy occurs when there is a chromosome missing from a pair. Turner syndrome is caused by a partial or complete deletion of the second X chromosome in a genetic female. This syndrome causes suppressed physical growth, and possibly future reproductive and other health issues (Boston Children’s Hospital, n.d.).
Prenatal testing can identify potential genetic disorders in the first or second trimester. Prenatal screening is done during the first trimester, and involves a blood test that checks fetal DNA matter found in the mother’s bloodstream for genetic mutations like trisomies or monosomies. Nuchal translucency screening (NTS) is an ultrasound that measures the thickness of the neck of the fetus, as more fluid buildup can indicate an increased risk of trisomies. An anatomical ultrasound scan is also typically done around week 20 of pregnancy to measure fetal growth and screen for malformations of the head, internal organs, or limbs (ACOG, 2023b). Diagnostic tests cannot typically be done until the second trimester, and include taking samples from the amniotic fluid (amniocentesis) or placenta (chorionic villus sampling). While screening tests are highly accurate at predicting the risk for specific genetic disorders or malformations, diagnostic tests are usually used to confirm the presence of a disorder. Taking into account other risk factors like the parents’ family history or genetic risks, and maternal age (pregnancy at over age 35 tends to increase certain risks), physicians may recommend less-invasive prenatal screening first, and then diagnostic testing if the screens indicate an increased risk (Johns Hopkins Medicine, 2021).
Genetic testing of parents prior to pregnancy and prenatal screening or diagnostics during pregnancy can help inform parents’ decision making and family planning. Some parents may decide to not have children or adopt children if they are genetic carriers for certain syndromes. Considering a diagnosis where an infant would likely suffer and have a short life expectancy after birth, some parents may opt to terminate a pregnancy. Sometimes when a diagnosis is discovered and the pregnancy is continued, parents can make preparations for services and support for both themselves and an infant with a disability. Still other parents may prefer to not know a diagnosis until birth, as it would not change their plans. Regardless of parents’ healthcare decisions, it is important that parents have access to accurate information regarding screening, diagnostic tests, and the potential impacts (ACOG, 2023b, Schneider, 2020).
Genetic screening also comes with ethical questions from a population perspective. Although substantial suffering and death from cruel diseases could potentially be prevented with genetic screening, emerging gene therapies, early medical intervention, and/or family planning, there is also the real risk of racism and ableism affecting genetic medicine. Historically, eugenics and racialized medicine have been used to justify atrocities: the Holocaust in Nazi Germany, forced sterilization of Black women and immigrants in the U.S., and other attempts at ethnic cleansing and harming or killing those with disabilities around the world. Even when genetic screening and counseling is done ethically, there is a risk of genetic information influencing a person’s social life, their potential for employment, and thus also their ability to get health insurance coverage. Genetic testing and prenatal screening therefore should always be voluntary and confidential, include education and counseling, and include tests that are of high quality and evidence-based (Schneider, 2020).


10.2: Mutual Inductance
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LEARNING OBJECTIVES
By the end of this section, you will be able to:
- Correlate two nearby circuits that carry time-varying currents with the emf induced in each circuit
- Describe examples in which mutual inductance may or may not be desirable
Inductance is the property of a device that tells us how effectively it induces an emf in another device. In other words, it is a physical quantity that expresses the effectiveness of a given device.
When two circuits carrying time-varying currents are close to one another, the magnetic flux through each circuit varies because of the changing current I in the other circuit. Consequently, an emf is induced in each circuit by the changing current in the other. This type of emf is therefore called a mutually induced emf , and the phenomenon that occurs is known as mutual inductance ( M ) . As an example, let’s consider two tightly wound coils (Figure \(\PageIndex{1}\)). Coils 1 and 2 have \(N_1\) and \(N_2\) turns and carry currents \(I_1\) and \(I_2\) respectively. The flux through a single turn of coil 2 produced by the magnetic field of the current in coil 1 is \(\Phi_{12}\), whereas the flux through a single turn of coil 1 due to the magnetic field of \(I_2\) is \(\Phi_{12}\).
The mutual inductance \(M_{21}\) of coil 2 with respect to coil 1 is the ratio of the flux through the \(N_2\) turns of coil 2 produced by the magnetic field of the current in coil 1, divided by that current, that is,
\[M_{21} = \dfrac{N_2\Phi_{21}}{I_1}. \label{12.24} \]
Similarly, the mutual inductance of coil 1 with respect to coil 2 is
\[M_{12} = \dfrac{N_1\Phi_{12}}{I_2}. \label{12.25} \]
Like capacitance, mutual inductance is a geometric quantity. It depends on the shapes and relative positions of the two coils, and it is independent of the currents in the coils. The SI unit for mutual inductance M is called the henry (H) in honor of Joseph Henry (1799–1878), an American scientist who discovered induced emf independently of Faraday. Thus, we have \(1 \, H = 1 \, V \cdot s/A\). From Equations \ref{12.24} and \ref{12.25}, we can show that \(M_{21} = M_{12}\), so we usually drop the subscripts associated with mutual inductance and write
\[M = \dfrac{N_2\Phi_{21}}{I_1} = \dfrac{N_1 \Phi_{12}}{I_2}.\label{14.3} \]
The emf developed in either coil is found by combining Faraday’s law and the definition of mutual inductance. Since \(N_2\Phi_{21}\) is the total flux through coil 2 due to \(I_1\), we obtain
\[\begin{align} \epsilon_2 &= - \dfrac{d}{dt} (N_2 \Phi_{21}) \\[4pt] &= - \dfrac{d}{dt} (MI_1) \\[4pt] & = - M\dfrac{dI_1}{dt} \label{14.4} \end{align} \]
where we have used the fact that \(M\) is a time-independent constant because the geometry is time-independent. Similarly, we have
\[\epsilon_1 = - M\dfrac{dI_2}{dt}. \label{14.5} \]
In Equation \ref{14.5}, we can see the significance of the earlier description of mutual inductance (\(M\)) as a geometric quantity. The value of \(M\) neatly encapsulates the physical properties of circuit elements and allows us to separate the physical layout of the circuit from the dynamic quantities, such as the emf and the current. Equation \ref{14.5} defines the mutual inductance in terms of properties in the circuit, whereas the previous definition of mutual inductance in Equation \ref{12.24} is defined in terms of the magnetic flux experienced, regardless of circuit elements. You should be careful when using Equations \ref{14.4} and \ref{14.4} because \(\epsilon_1\) and \(\epsilon_2\) do not necessarily represent the total emfs in the respective coils. Each coil can also have an emf induced in it because of its self-inductance (self-inductance will be discussed in more detail in a later section).
A large mutual inductance M may or may not be desirable. We want a transformer to have a large mutual inductance. But an appliance, such as an electric clothes dryer, can induce a dangerous emf on its metal case if the mutual inductance between its coils and the case is large. One way to reduce mutual inductance is to counter-wind coils to cancel the magnetic field produced (Figure \(\PageIndex{2}\)).
Digital signal processing is another example in which mutual inductance is reduced by counter-winding coils. The rapid on/off emf representing 1s and 0s in a digital circuit creates a complex time-dependent magnetic field. An emf can be generated in neighboring conductors. If that conductor is also carrying a digital signal, the induced emf may be large enough to switch 1s and 0s, with consequences ranging from inconvenient to disastrous.
Example \(\PageIndex{1}\): Mutual Inductance
Figure \(\PageIndex{3}\) shows a coil of \(N_2\) turns and radius \(R_2\) surrounding a long solenoid of length \(l_1\), radius \(R_1\), and \(N_1\) turns.
- What is the mutual inductance of the two coils?
- If \(N_1 = 500 \, turns, \, N_2 = 10 \, turns, \, R_1 = 3.10 \, cm, \, l_1 = 75.0 \, cm\), and the current in the solenoid is changing at a rate of 200 A/s, what is the emf induced in the surrounding coil?
Strategy
There is no magnetic field outside the solenoid, and the field inside has magnitude \(B_1 = \mu_0(N_1/l_1)I_1\) and is directed parallel to the solenoid’s axis. We can use this magnetic field to find the magnetic flux through the surrounding coil and then use this flux to calculate the mutual inductance for part (a), using Equation \ref{14.3}. We solve part (b) by calculating the mutual inductance from the given quantities and using Equation \ref{14.4} to calculate the induced emf.
Solution
- The magnetic flux \(\Phi_{21}\) through the surrounding coil is \[\begin{align*} \Phi_{21} &= B_1 \pi R_1^2 \\[4pt] &= \dfrac{\mu_0 N_1I_1}{l_1}\pi R_1^2. \end{align*} \] Now from Equation \ref{14.3}, the mutual inductance is \[\begin{align*} M &= \dfrac{N_2\Phi_{21}}{I_1} \\[4pt] &= \left(\dfrac{N_2}{I_1}\right)\left(\dfrac{\mu_0N_1I_1}{l_1}\right) \pi R_1^2 \nonumber \\[4pt] &= \dfrac{\mu_0N_1N_2 \pi R_1^2}{l_1}.\end{align*} \nonumber \]
- Using the previous expression and the given values, the mutual inductance is \[\begin{align*} M &= \dfrac{(4\pi \times 10^{-7} \, T \cdot m/A)(500)(10)\pi (0.0310 \, m)^2}{0.750 \, m} \\[4pt] &=2.53 \times 10^{-5} \, H. \nonumber \end{align*} \nonumber \] Thus, from Equation \ref{14.4}, the emf induced in the surrounding coil is \[\begin{align*} \epsilon_2 &= - M\dfrac{dI_1}{dt} \\[4pt] &= - (2.53 \times 10^{-5} H)(200 \, A/s) \\[4pt] &= - 5.06 \times 10^{-3}V. \end{align*} \nonumber \]
Significance
Notice that M in part (a) is independent of the radius \(R_2\) of the surrounding coil because the solenoid’s magnetic field is confined to its interior. In principle, we can also calculate M by finding the magnetic flux through the solenoid produced by the current in the surrounding coil. This approach is much more difficult because \(\Phi_{12}\) is so complicated. However, since \(M_{12} = M_{21}\), we do know the result of this calculation.
Exercise \(\PageIndex{1}\)
A current \(I(t) = (5.0 \, A) \, sin \, ((120\pi \, rad/s)t)\) flows through the solenoid of part (b) of Example \(\PageIndex{1}\). What is the maximum emf induced in the surrounding coil?
Solution
\(4.77 \times 10^{-2} \, V\)


4.3: Inner Product and Euclidean Norm
The inner product \((x,y)\) between vectors \(x\) and \(y\) is a scalar consisting of the following sum of products:
\[(x,y) = x_1y_1+x_2y_2+x_3y_3+⋯+x_ny_n \nonumber \]
This definition seems so arbitrary that we wonder what uses it could possibly have. We will show that the inner product has three main uses:
- computing length or “norm”,
- finding angles between vectors and checking for “orthogonality”, and
- computing the “component of one vector along another” (projection).
Since the inner product is so useful, we need to know what algebraic operations are permitted when we are working with inner products. The following are some properties of the inner product. Given \(x,y,z\;∈\;\mathbb{R}^n\) and \(a\;∈\;\mathbb{R}\),
- \((x,y)=(y,x)\);
- \((ax,y)=a(x,y)=(x,ay)\); and
- \((x,y+z)=(x,y)+(x,z)\).
Exercise \(\PageIndex{1}\)
Prove the three preceding properties by using the definition of inner product. Is the equation \(x(y,z)=(x,y)z\) also a true property?
Euclidean Norm
Sometimes we want to measure the length of a vector, namely, the distance from the origin to the point specified by the vector's coordinates. A vector's length is called the norm of the vector. Recall from Euclidean geometry that the distance between two points is the square root of the sum of the squares of the distances in each dimension. Since we are measuring from the origin, this implies that the norm of the vector x is
\[||x||=\sqrt{x^2_1+x^2_2+...+x^2_n} \nonumber \]
Notice the use of the double vertical bars to indicate the norm. An equivalent definition of the norm, and of the norm squared, can be given in terms of the inner product:
\[||x||=\sqrt{(x,x)} \nonumber \]
or
\[||x||^2=(x,x) \nonumber \]
The Euclidean norm of the vector
\(x=\begin{bmatrix} 1\\3\\5\\−2 \end{bmatrix}\)
is \(||x||=\sqrt{1^2+3^2+5^2+(−2)^2}=\sqrt{39}\)
An important property of the norm and scalar product is that, for any \(x\;∈\;\mathbb{R}^n\) and \(a\;∈\;\mathbb{R}\),
\[||ax||=|a|||x|| \nonumber \]
So we can take a scalar multiplier outside of the norm if we take its absolute value.
Exercise \(\PageIndex{2}\)
Prove \(||ax||=|a|||x||\)
Cauchy-Schwarz Inequality
Inequalities can be useful engineering tools. They can often be used to find the best possible performance of a system, thereby telling you when to quit trying to make improvements (or proving to your boss that it can't be done any better). The most fundamental inequality in linear algebra is the Cauchy-Schwarz inequality. This inequality says that the inner product between two vectors \(x\) and \(y\) is less than or equal (in absolute value) to the norm of \(x\) times the norm of \(y\), with equality if and only if \(y=αx\):
\[|(x,y)|≤||x||\;||y|| \nonumber \]
To prove this theorem, we construct a third vector \(z=λx−y\) and measure its norm squared:
\[||λx−y||^2=(λx−y,λx−y)=λ^2||x||^2−2λ(x,y)+||y||^2≥0 \nonumber \]
So we have a polynomial in \(λ\) that is always greater than or equal to 0 (because every norm squared is greater than or equal to 0). Let's assume that \(x\) and \(y\) are given and minimize this norm squared with respect to \(λ\). To do this, we take the derivative with respect to \(λ\) and equate it to 0:
\[2λ||x||^2−2(x,y)=0\;⇒\;λ=\frac{(x,y)} {||x||^2} \nonumber \]
When this solution is substituted into the formula for the norm squared in Equation\(\PageIndex{7}\), we obtain
\[\begin{bmatrix} \frac {(x,y)} {||x||^2}\end{bmatrix}^2\;||x||^2−\frac {2(x,y)} {||x||^2} (x,y)+||y||^2≥0 \nonumber \]
which simplifies to
\[-\frac {(x,y)^2} {||x||^2} + ||y||^2 ≥ 0 ⇒ (x,y)^2 ≤ ||x||^2||y||^2 \nonumber \]
The proof of the Cauchy-Schwarz inequality is completed by taking the positive square root on both sides of the above Equation. When \(y=αx\), then
\[(x,y)^2=(x,αx)^2=[|α|(x,x)]^2=(|α|||x||^2)^2 \nonumber \]
\[=(|α|^2||x||^2)||x||^2 \nonumber \]
\[=(αx,αx)||x||^2 \nonumber \]
\[=(y,y)||x||^2 \nonumber \]
\[=||y||^2||x|^|2 \nonumber \]
which shows that equality holds in Equation \(\PageIndex{6}\) when y is a scalar multiple of x.
Exercise \(\PageIndex{3}\)
Use the Cauchy-Schwarz inequality to prove the triangle inequality, which states
\(||x+y||≤||x||+||y||\)
Explain why this is called the triangle inequality.


1.7: Exercise- Plotting ERP Scalp Maps
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As a last step in our quick tour of EEGLAB and ERPLAB, we’re going to plot a topographic map (aka scalp map ) of the ERPs.
Select ERPLAB > Plot ERP > Plot Scalp maps , which will bring up a dialog box that lets you control the plotting parameters. Enter 1 i nto the text box at the upper left labeled Bin(s) to plot (so that it plots bin 1) and enter 400 into the text box labeled Latencies to plot in ms (so that it plots the voltage at 400 ms). Then check the box labeled display color scale box near the right edge of the window. Finally, click the PLOT button in the lower right corner. You should see something like Screenshot 1.11.
Each little dot on the head is one of the electrode sites. The coloring indicates the voltage at each point on the scalp (using interpolation to fill in the values between electrodes). The scale along the right side indicates what amplitude (in microvolts) is indicated by a particular color.


9.1: Introduction to Electricity
The image of American politician and scientist Benjamin Franklin (1706–1790) flying a kite in a thunderstorm is familiar to every schoolchild. (See Figure .) In this experiment, Franklin demonstrated a connection between lightning and static electricity . Sparks were drawn from a key hung on a kite string during an electrical storm. These sparks were like those produced by static electricity, such as the spark that jumps from your finger to a metal doorknob after you walk across a wool carpet. What Franklin demonstrated in his dangerous experiment was a connection between phenomena on two different scales: one the grand power of an electrical storm, the other an effect of more human proportions. Connections like this one reveal the underlying unity of the laws of nature, an aspect we humans find particularly appealing.
Our understanding of electricity and the range of phenomena that are electrical in nature has vastly expanded since those early days. For example, atomic and molecular interactions, such as the forces of friction, cohesion, and adhesion, are now known to be manifestations of the electromagnetic force . Static electricity is just one aspect of the electromagnetic force, which also includes moving electricity and magnetism. Furthermore, All the macroscopic forces that we experience directly, such as the sensations of touch and the tension in a rope, are due to the electromagnetic force, one of the four fundamental forces in nature. The gravitational force, another fundamental force, is actually sensed through the electromagnetic interaction of molecules, such as between those in our feet and those on the top of a bathroom scale. (The other two fundamental forces, the strong nuclear force and the weak nuclear force, cannot be sensed on the human scale.)
In this chapter, we will develop the concepts of electric fields and electric potential (also known as voltage ), which are used to describe electric force and electrical energy. Consider, for example, great amounts of electrical energy stored in batteries or transmitted cross-country through power lines, or consider electrical signals sent in our nervous systems at molecular levels, with ions crossing cell membranes and transferring information, or the role of electricity in our household appliances and devices, including your laptop and smartphones. We will look at the laws that help us describe electric circuits and begin to explore some of the many applications of electricity.
Glossary
- static electricity
- a buildup of electric charge on the surface of an object
- electromagnetic force
- one of the four fundamental forces of nature; the electromagnetic force consists of static electricity, moving electricity and magnetism


9.4: Fermentation
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- Ying Liu
- City College of San Francisco
Learning Objectives
- Define fermentation and explain why it does not require oxygen
- Describe the fermentation pathways and their end products and give examples of microorganisms that use these pathways
- Compare and contrast fermentation and anaerobic respiration
Many cells are unable to carry out respiration because of one or more of the following circumstances:
- The cell lacks a sufficient amount of any appropriate, inorganic, final electron acceptor to carry out cellular respiration.
- The cell lacks genes to make appropriate complexes and electron carriers in the electron transport system.
- The cell lacks genes to make one or more enzymes in the Krebs cycle.
Whereas lack of an appropriate inorganic final electron acceptor is environmentally dependent, the other two conditions are genetically determined. Thus, many prokaryotes, including members of the clinically important genus Streptococcus , are permanently incapable of respiration, even in the presence of oxygen. Conversely, many prokaryotes are facultative, meaning that, should the environmental conditions change to provide an appropriate inorganic final electron acceptor for respiration, organisms containing all the genes required to do so will switch to cellular respiration for glucose metabolism because respiration allows for much greater ATP production per glucose molecule.
If respiration does not occur, NADH must be reoxidized to NAD + for reuse as an electron carrier for glycolysis, the cell’s only mechanism for producing any ATP, to continue. Some living systems use an organic molecule (commonly pyruvate) as a final electron acceptor through a process called fermentation. Fermentation does not involve an electron transport system and does not directly produce any additional ATP beyond that produced during glycolysis by substrate-level phosphorylation. Organisms carrying out fermentation, called fermenters, produce a maximum of two ATP molecules per glucose during glycolysis. Table \(\PageIndex{1}\) compares the final electron acceptors and methods of ATP synthesis in aerobic respiration, anaerobic respiration, and fermentation. Note that the number of ATP molecules shown for glycolysis assumes the Embden-Meyerhof-Parnas pathway. The number of ATP molecules made by substrate-level phosphorylation (SLP) versus oxidative phosphorylation (OP) are indicated.
| Type of Metabolism | Example | Final Electron Acceptor | Pathways Involved in ATP Synthesis (Type of Phosphorylation) | Maximum Yield of ATP Molecules |
|---|---|---|---|---|
| Aerobic respiration | Pseudomonas aeruginosa | \(\ce{O2}\) |
EMP glycolysis (SLP) Krebs cycle (SLP) Electron transport and chemiosmosis (OP): |
2 2 34 |
| Total | 38 | |||
| Anaerobic respiration | Paracoccus denitrificans |
\(\ce{NO3-}\), \(\ce{SO4^{-2}}\), \(\ce{Fe^{+3}}\), \(\ce{CO2}\), other inorganics |
EMP glycolysis (SLP) Krebs cycle (SLP) Electron transport and chemiosmosis (OP): |
2 2 1–32 |
| Total | 5 – 36 | |||
| Fermentation | Candida albicans |
Organics (usually pyruvate) |
EMP glycolysis (SLP) Fermentation |
2 0 |
| Total | 2 |
Microbial fermentation processes have been manipulated by humans and are used extensively in the production of various foods and other commercial products, including pharmaceuticals. Microbial fermentation can also be useful for identifying microbes for diagnostic purposes.
Fermentation by some bacteria, like those in yogurt and other soured food products, and by animals in muscles during oxygen depletion, is lactic acid fermentation. The chemical reaction of lactic acid fermentation is as follows:
\[\ce{Pyruvate + NADH \leftrightarrow lactic\: acid + NAD+}\]
Bacteria of several gram-positive genera, including Lactobacillus , Leuconostoc , and Streptococcus , are collectively known as the lactic acid bacteria (LAB), and various strains are important in food production. During yogurt and cheese production, the highly acidic environment generated by lactic acid fermentation denatures proteins contained in milk, causing it to solidify. When lactic acid is the only fermentation product, the process is said to be homolactic fermentation; such is the case for Lactobacillus delbrueckii and S. thermophiles used in yogurt production. However, many bacteria perform heterolactic fermentation, producing a mixture of lactic acid, ethanol and/or acetic acid, and CO 2 as a result, because of their use of the branched pentose phosphate pathway instead of the EMP pathway for glycolysis. One important heterolactic fermenter is Leuconostoc mesenteroides , which is used for souring vegetables like cucumbers and cabbage, producing pickles and sauerkraut, respectively.
Lactic acid bacteria are also important medically. The production of low pH environments within the body inhibits the establishment and growth of pathogens in these areas. For example, the vaginal microbiota is composed largely of lactic acid bacteria, but when these bacteria are reduced, yeast can proliferate, causing a yeast infection. Additionally, lactic acid bacteria are important in maintaining the health of the gastrointestinal tract and, as such, are the primary component of probiotics.
Another familiar fermentation process is alcohol fermentation, which produces ethanol. The ethanol fermentation reaction is shown in Figure \(\PageIndex{1}\). In the first reaction, the enzyme pyruvate decarboxylase removes a carboxyl group from pyruvate, releasing CO 2 gas while producing the two-carbon molecule acetaldehyde. The second reaction, catalyzed by the enzyme alcohol dehydrogenase, transfers an electron from NADH to acetaldehyde, producing ethanol and NAD + . The ethanol fermentation of pyruvate by the yeast Saccharomyces cerevisiae is used in the production of alcoholic beverages and also makes bread products rise due to CO 2 production. Outside of the food industry, ethanol fermentation of plant products is important in biofuel production.
Beyond lactic acid fermentation and alcohol fermentation, many other fermentation methods occur in prokaryotes, all for the purpose of ensuring an adequate supply of NAD + for glycolysis (Table \(\PageIndex{2}\)). Without these pathways, glycolysis would not occur and no ATP would be harvested from the breakdown of glucose. It should be noted that most forms of fermentation besides homolactic fermentation produce gas, commonly CO 2 and/or hydrogen gas. Many of these different types of fermentation pathways are also used in food production and each results in the production of different organic acids, contributing to the unique flavor of a particular fermented food product. The propionic acid produced during propionic acid fermentation contributes to the distinctive flavor of Swiss cheese, for example.
Several fermentation products are important commercially outside of the food industry. For example, chemical solvents such as acetone and butanol are produced during acetone-butanol-ethanol fermentation. Complex organic pharmaceutical compounds used in antibiotics (e.g., penicillin), vaccines, and vitamins are produced through mixed acid fermentation. Fermentation products are used in the laboratory to differentiate various bacteria for diagnostic purposes. For example, enteric bacteria are known for their ability to perform mixed acid fermentation, reducing the pH, which can be detected using a pH indicator. Similarly, the bacterial production of acetoin during butanediol fermentation can also be detected. Gas production from fermentation can also be seen in an inverted Durham tube that traps produced gas in a broth culture.
Microbes can also be differentiated according to the substrates they can ferment. For example, E. coli can ferment lactose, forming gas, whereas some of its close gram-negative relatives cannot. The ability to ferment the sugar alcohol sorbitol is used to identify the pathogenic enterohemorrhagic O157:H7 strain of E. coli because, unlike other E. coli strains, it is unable to ferment sorbitol. Last, mannitol fermentation differentiates the mannitol-fermenting Staphylococcus aureus from other non–mannitol-fermenting staphylococci.
| Pathway | End Products | Example Microbes | Commercial Products |
|---|---|---|---|
| Acetone-butanol-ethanol | Acetone, butanol, ethanol, CO 2 | Clostridium acetobutylicum | Commercial solvents, gasoline alternative |
| Alcohol | Ethanol, CO 2 | Candida, Saccharomyces | Beer, bread |
| Butanediol | Formic and lactic acid; ethanol; acetoin; 2,3 butanediol; CO 2 ; hydrogen gas | Klebsiella, Enterobacter | Chardonnay wine |
| Butyric acid | Butyric acid, CO 2 , hydrogen gas | Clostridium butyricum | Butter |
| Lactic acid | Lactic acid | Streptococcus, Lactobacillus | Sauerkraut, yogurt, cheese |
| Mixed acid | Acetic, formic, lactic, and succinic acids; ethanol, CO 2 , hydrogen gas | Escherichia, Shigella | Vinegar, cosmetics, pharmaceuticals |
| Propionic acid | Acetic acid, propionic acid, CO 2 | Propionibacterium, Bifidobacterium | Swiss cheese |
Exercise \(\PageIndex{1}\)
When would a metabolically versatile microbe perform fermentation rather than cellular respiration?
Identifying Bacteria by Using API Test Panels
Identification of a microbial isolate is essential for the proper diagnosis and appropriate treatment of patients. Scientists have developed techniques that identify bacteria according to their biochemical characteristics. Typically, they either examine the use of specific carbon sources as substrates for fermentation or other metabolic reactions, or they identify fermentation products or specific enzymes present in reactions. In the past, microbiologists have used individual test tubes and plates to conduct biochemical testing. However, scientists, especially those in clinical laboratories, now more frequently use plastic, disposable, multitest panels that contain a number of miniature reaction tubes, each typically including a specific substrate and pH indicator. After inoculation of the test panel with a small sample of the microbe in question and incubation, scientists can compare the results to a database that includes the expected results for specific biochemical reactions for known microbes, thus enabling rapid identification of a sample microbe. These test panels have allowed scientists to reduce costs while improving efficiency and reproducibility by performing a larger number of tests simultaneously.
Many commercial, miniaturized biochemical test panels cover a number of clinically important groups of bacteria and yeasts. One of the earliest and most popular test panels is the Analytical Profile Index (API) panel invented in the 1970s. Once some basic laboratory characterization of a given strain has been performed, such as determining the strain’s Gram morphology, an appropriate test strip that contains 10 to 20 different biochemical tests for differentiating strains within that microbial group can be used. Currently, the various API strips can be used to quickly and easily identify more than 600 species of bacteria, both aerobic and anaerobic, and approximately 100 different types of yeasts. Based on the colors of the reactions when metabolic end products are present, due to the presence of pH indicators, a metabolic profile is created from the results (Figure \(\PageIndex{2}\)). Microbiologists can then compare the sample’s profile to the database to identify the specific microbe.
Clinical Focus: Part 2
Many of Hannah’s symptoms are consistent with several different infections, including influenza and pneumonia. However, her sluggish reflexes along with her light sensitivity and stiff neck suggest some possible involvement of the central nervous system, perhaps indicating meningitis. Meningitis is an infection of the cerebrospinal fluid (CSF) around the brain and spinal cord that causes inflammation of the meninges, the protective layers covering the brain. Meningitis can be caused by viruses, bacteria, or fungi. Although all forms of meningitis are serious, bacterial meningitis is particularly serious. Bacterial meningitis may be caused by several different bacteria, but the bacterium Neisseria meningitidis , a gram-negative, bean-shaped diplococcus, is a common cause and leads to death within 1 to 2 days in 5% to 10% of patients.
Given the potential seriousness of Hannah’s conditions, her physician advised her parents to take her to the hospital in the Gambian capital of Banjul and there have her tested and treated for possible meningitis. After a 3-hour drive to the hospital, Hannah was immediately admitted. Physicians took a blood sample and performed a lumbar puncture to test her CSF. They also immediately started her on a course of the antibiotic ceftriaxone, the drug of choice for treatment of meningitis caused by N. meningitidis , without waiting for laboratory test results.
Exercise \(\PageIndex{2}\)
- How might biochemical testing be used to confirm the identity of N. meningitidis ?
- Why did Hannah’s doctors decide to administer antibiotics without waiting for the test results?
Key Concepts and Summary
- Fermentation uses an organic molecule as a final electron acceptor to regenerate NAD + from NADH so that glycolysis can continue.
- Fermentation does not involve an electron transport system, and no ATP is made by the fermentation process directly. Fermenters make very little ATP—only two ATP molecules per glucose molecule during glycolysis.
- Microbial fermentation processes have been used for the production of foods and pharmaceuticals, and for the identification of microbes.
- During lactic acid fermentation, pyruvate accepts electrons from NADH and is reduced to lactic acid. Microbes performing homolactic fermentation produce only lactic acid as the fermentation product; microbes performing heterolactic fermentation produce a mixture of lactic acid, ethanol and/or acetic acid, and CO 2 .
- Lactic acid production by the normal microbiota prevents growth of pathogens in certain body regions and is important for the health of the gastrointestinal tract.
- During ethanol fermentation, pyruvate is first decarboxylated (releasing CO 2 ) to acetaldehyde, which then accepts electrons from NADH, reducing acetaldehyde to ethanol. Ethanol fermentation is used for the production of alcoholic beverages, for making bread products rise, and for biofuel production.
- Fermentation products of pathways (e.g., propionic acid fermentation) provide distinctive flavors to food products. Fermentation is used to produce chemical solvents (acetone-butanol-ethanol fermentation) and pharmaceuticals (mixed acid fermentation).
- Specific types of microbes may be distinguished by their fermentation pathways and products. Microbes may also be differentiated according to the substrates they are able to ferment.


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1: An Invisible World
Week 1
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1.2: Systematic Approach
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BIOL 342: The New Plagues - New and Ancient Infectious Diseases Threatening World Health (Hughes)
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1: An Invisible World
1.2: Systematic Approach
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5.7: The Doppler Effect
By the end of this section, you will be able to:
- Explain why the spectral lines of photons we observe from an object will change as a result of the object’s motion toward or away from us
- Describe how we can use the Doppler effect to deduce how fast astronomical objects are moving through space
The last two sections introduced you to many new concepts, and we hope that through those, you have seen one major idea emerge. Astronomers can learn about the elements in stars and galaxies by decoding the information in their spectral lines. There is a complicating factor in learning how to decode the message of starlight, however. If a star is moving toward or away from us, its lines will be in a slightly different place in the spectrum from where they would be in a star at rest. And most objects in the universe do have some motion relative to the Sun.
Motion Affects Waves
In 1842, Christian Doppler first measured the effect of motion on waves by hiring a group of musicians to play on an open railroad car as it was moving along the track. He then applied what he learned to all waves, including light, and pointed out that if a light source is approaching or receding from the observer, the light waves will be, respectively, crowded more closely together or spread out. The general principle, now known as the Doppler effect , is illustrated in Figure 5.22.
In part (a) of the figure, the light source (S) is at rest with respect to the observer. The source gives off a series of waves, whose crests we have labeled 1, 2, 3, and 4. The light waves spread out evenly in all directions, like the ripples from a splash in a pond. The crests are separated by a distance, λ, where λ is the wavelength. The observer, who happens to be located in the direction of the bottom of the image, sees the light waves coming nice and evenly, one wavelength apart. Observers located anywhere else would see the same thing.
On the other hand, if the source of light is moving with respect to the observer, as seen in part (b), the situation is more complicated. Between the time one crest is emitted and the next one is ready to come out, the source has moved a bit, toward the bottom of the page. From the point of view of observer A , this motion of the source has decreased the distance between crests—it’s squeezing the crests together, this observer might say.
In part (b), we show the situation from the perspective of three observers. The source is seen in four positions, S 1 , S 2 , S 3 , and S 4 , each corresponding to the emission of one wave crest. To observer A , the waves seem to follow one another more closely, at a decreased wavelength and thus increased frequency. (Remember, all light waves travel at the speed of light through empty space, no matter what. This means that motion cannot affect the speed, but only the wavelength and the frequency. As the wavelength decreases, the frequency must increase. If the waves are shorter, more will be able to move by during each second.)
The situation is not the same for other observers. Let’s look at the situation from the point of view of observer C , located opposite observer A in the figure. For her, the source is moving away from her location. As a result, the waves are not squeezed together but instead are spread out by the motion of the source. The crests arrive with an increased wavelength and decreased frequency. To observer B , in a direction at right angles to the motion of the source, no effect is observed. The wavelength and frequency remain the same as they were in part (a) of the figure.
We can see from this illustration that the Doppler effect is produced only by a motion toward or away from the observer, a motion called radial velocity . Sideways motion does not produce such an effect. Observers between A and B would observe some shortening of the light waves for that part of the motion of the source that is along their line of sight. Observers between B and C would observe lengthening of the light waves that are along their line of sight.
You may have heard the Doppler effect with sound waves. When a train whistle or police siren approaches you and then moves away, you will notice a decrease in the pitch (which is how human senses interpret sound wave frequency) of the sound waves. Compared to the waves at rest, they have changed from slightly more frequent when coming toward you, to slightly less frequent when moving away from you.
A nice example of this change in the sound of a train whistle can be heard at the end of the classic Beach Boys song “Caroline, No” on their album Pet Sounds . To hear this sound, go to this YouTube version of the song. The sound of the train begins at approximately 2:20.
Color Shifts
When the source of waves moves toward you, the wavelength decreases a bit. If the waves involved are visible light, then the colors of the light change slightly. As wavelength decreases, they shift toward the blue end of the spectrum: astronomers call this a blueshift (since the end of the spectrum is really violet, the term should probably be violetshift , but blue is a more common color). When the source moves away from you and the wavelength gets longer, we call the change in colors a redshift . Because the Doppler effect was first used with visible light in astronomy, the terms “ blueshift ” and “ redshift ” became well established. Today, astronomers use these words to describe changes in the wavelengths of radio waves or X-rays as comfortably as they use them to describe changes in visible light.
The greater the motion toward or away from us, the greater the Doppler shift. If the relative motion is entirely along the line of sight, the formula for the Doppler shift of light is
where λ is the wavelength emitted by the source, Δλ is the difference between λ and the wavelength measured by the observer, c is the speed of light, and v is the relative speed of the observer and the source in the line of sight. The variable v is counted as positive if the velocity is one of recession, and negative if it is one of approach. Solving this equation for the velocity, we find v = c × Δλ/λ.
Click-and-drag the object emitting waves or the object receiving waves in this simulator to experiment with the Doppler effect yourself. The plots along the top show how the detected wavelengths change if the objects are approaching each other or moving further apart.
If a star approaches or recedes from us, the wavelengths of light in its continuous spectrum appear shortened or lengthened, respectively, as do those of the dark lines. However, unless its speed is tens of thousands of kilometers per second, the star does not appear noticeably bluer or redder than normal. The Doppler shift is thus not easily detected in a continuous spectrum and cannot be measured accurately in such a spectrum. The wavelengths of the absorption lines can be measured accurately, however, and their Doppler shift is relatively simple to detect.
We can use the Doppler effect equation to calculate the radial velocity of an object if we know three things: the speed of light, the original (unshifted) wavelength of the light emitted, and the difference between the wavelength of the emitted light and the wavelength we observe. For particular absorption or emission lines, we usually know exactly what wavelength the line has in our laboratories on Earth, where the source of light is not moving. We can measure the new wavelength with our instruments at the telescope, and so we know the difference in wavelength due to Doppler shifting. Since the speed of light is a universal constant, we can then calculate the radial velocity of the star.
A particular emission line of hydrogen is originally emitted with a wavelength of 656.3 nm from a gas cloud. At our telescope, we observe the wavelength of the emission line to be 656.6 nm. How fast is this gas cloud moving toward or away from Earth?
Solution
Because the light is shifted to a longer wavelength (redshifted), we know this gas cloud is moving away from us. The speed can be calculated using the Doppler shift formula:
Suppose a spectral line of hydrogen, normally at 500 nm, is observed in the spectrum of a star to be at 500.1 nm. How fast is the star moving toward or away from Earth?
- Answer
-
Because the light is shifted to a longer wavelength, the star is moving away from us:
Its speed is
60,000 m/s.
You may now be asking: if all the stars are moving and motion changes the wavelength of each spectral line, won’t this be a disaster for astronomers trying to figure out what elements are present in the stars? After all, it is the precise wavelength (or color) that tells astronomers which lines belong to which element. And we first measure these wavelengths in containers of gas in our laboratories, which are not moving. If every line in a star’s spectrum is now shifted by its motion to a different wavelength (color), how can we be sure which lines and which elements we are looking at in a star whose speed we do not know?
Take heart. This situation sounds worse than it really is. Astronomers rarely judge the presence of an element in an astronomical object by a single line. It is the pattern of lines unique to hydrogen or calcium that enables us to determine that those elements are part of the star or galaxy we are observing. The Doppler effect does not change the pattern of lines from a given element—it only shifts the whole pattern slightly toward redder or bluer wavelengths. The shifted pattern is still quite easy to recognize. Best of all, when we do recognize a familiar element’s pattern, we get a bonus: the amount the pattern is shifted can enable us to determine the speed of the objects in our line of sight.
The training of astronomers includes much work on learning to decode light (and other electromagnetic radiation). A skillful “decoder” can learn the temperature of a star, what elements are in it, and even its speed in a direction toward us or away from us. That’s really an impressive amount of information for stars that are light-years away.


17: Leslie Marmon Silko (1948- )
Thumbnail: Headliner Leslie Marmon Silko at the banquet reading. (CC BY-SA 2.0; Uche Ogbuji via Wikipedia )
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Leslie Marmon Silko (1948- ) Yellow Woman Native American Contemporary Literature Leslie Marmon Silko was born in Albuquerque, New Mexico, but was raised in Laguna Pueblo. She is a talented poet and prose writer, whose work incorporates elements of Native American storytelling traditions. She studied English at the University of New Mexico, and graduated with honors. After her graduation, she published her first story, "Tony's Song." She briefly studied law, but left the program to pursue a graduate degree in English. In 1974, she published several stories in Kenneth Rosen's anthology, The Man to Send Rain Clouds: Contemporary Stories by American Indians . Her first novel, Ceremony , a World War II veteran's attempts to find peace after the war, was published in 1977, to critical acclaim. The novel led to Silko being awarded a MacArthur Fellowship in 1981. In her writing, Silko commonly addresses ideas of healing and reconciling conflicts (cultural, spiritual, internal). "Yellow Woman" was first published in The Man to Send Rain Clouds: Contemporary Stories by American Indians . It is one of her most commonly anthologized pieces. In the story, Silko explores the Laguna tradition of Yellow Woman, who is often abducted, taken to the spirit world, and later returns with a great power that helps her people. Whether or not the characters from the story are Yellow Woman and other spirits is something that Silko does not clarify. The uncertainty is a compelling aspect of the story. Consider while reading:
Written by Laura Ng
Thumbnail: Headliner Leslie Marmon Silko at the banquet reading. (CC BY-SA 2.0; Uche Ogbuji via Wikipedia )


13.5: Conformal interpretation
Let us give another interpretation of the h-distance.
Consider the h-plane with the unit circle centered at \(O\) as the absolute. Fix a point \(P\) and let \(Q\) be another point in the h-plane. Set \(x = PQ\) and \(y = PQ_h\) . Then
\[\lim_{x\to 0} \dfrac{y}{x} = \dfrac{2}{1-OP^2}.\]
The above formula tells us that the h-distance from \(P\) to a nearby point \(Q\) is almost proportional to the Euclidean distance with the coefficient \(\dfrac{2}{1-OP^2}\) . The value \(\lambda(P)=\dfrac{2}{1-OP^2}\) is called the conformal factor of the h-metric.
The value \(\dfrac{1}{\lambda(P)}=\dfrac{1}{2} \cdot (1-OP^2)\) can be interpreted as the speed limit at the given point \(P\) . In this case the h-distance is the minimal time needed to travel from one point of the h-plane to another point.
- Proof
-
If \(P=O\) , then according to Lemma 12.3.2
\[\dfrac{y}{x}=\dfrac{\ln \dfrac{1+x}{1-x}}{x}\to 2\]
as \(x\to0\) .
If \(P\ne O\) , let \(Z\) denotes the inverse of \(P\) in the absolute. Suppose that \(\Gamma\) denotes the circle with the center \(Z\) perpendicular to the absolute.
According to the main observation ( Theorem 12.3.1 ) and Lemma 12.3.1 , the inversion in \(\Gamma\) is a motion of the h-plane which sends \(P\) to \(O\) . In particular, if \(Q'\) denotes the inverse of \(Q\) in \(\Gamma\) , then \(OQ'_h=PQ_h\) .
Set \(x'=OQ'\) . According to Lemma 10.1.1 ,
\(\dfrac{x'}{x}=\dfrac{OZ}{ZQ}.\)
Since \(Z\) is the inverse of \(P\) in the absolute, we have that \(PO\cdot OZ=1\) . Therefore,
as \(x \to 0\) .
According to 13.5.1, \(\dfrac{y}{x'} \to 2\) as \(x' \to 0\) . Therefore
as \(x \to 0\) .
Here is an application of the lemma above.
The circumference of an h-circle of the h-radius \(r\) is
\(2 \cdot \pi \cdot \sinh r,\)
where \(\sinh r\) denotes the hyperbolic sine of \(r\) ; that is,
\(\sinh r := \dfrac{e^r-e^{-r}}{2}.\)
Before we proceed with the proof, let us discuss the same problem in the Euclidean plane.
The circumference of a circle in the Euclidean plane can be defined as the limit of perimeters of regular \(n\) -gons inscribed in the circle as \(n \to \infty\) .
Namely, let us fix \(r>0\) . Given a positive integer \(n\) , consider \(\triangle AOB\) such that \(\measuredangle AOB=\dfrac{2\cdot\pi}{n}\) and \(OA=OB=r\) . Set \(x_n=AB\) . Note that \(x_n\) is the side of a regular \(n\) -gon inscribed in the circle of radius \(r\) . Therefore, the perimeter of the \(n\) -gon is \(n\cdot x_n\) .
The circumference of the circle with the radius \(r\) might be defined as the limit
\[\lim_{n\to\infty} n\cdot x_n=2\cdot\pi\cdot r.\]
(This limit can be taken as the definition of \(\pi\) .)
In the following proof, we repeat the same construction in the h-plane.
- Proof
-
Without loss of generality, we can assume that the center \(O\) of the circle is the center of the absolute.
By Lemma 12.3.2 , the h-circle with the h-radius \(r\) is the Euclidean circle with the center \(O\) and the radius
\(a=\dfrac{e^r-1}{e^r+1}.\)
Let \(x_n\) and \(y_n\) denote the side lengths of the regular \(n\) -gons inscribed in the circle in the Euclidean and hyperbolic plane respectively.
Note that \(x_n\to0\) as \(n\to\infty\) . By Lemma \(\PageIndecx{1}\),
\(\lim_{n\to\infty} \dfrac{y_n}{x_n} = \dfrac{2}{1-a^2}.\)
Applying 13.5.2, we get that the circumference of the h-circle can be found the following way:
\(\begin{array} {rcl} {\lim_{n \to \infty} n \cdot y_n} & = & {\dfrac{2}{1 - a^2} \cdot \lim_{n \to \infty} n \cdot x_n =} \\ {} & = & {\dfrac{4 \cdot \pi \cdot a}{1 - a^2} =} \\ {} & = & {\dfrac{4 \cdot \pi \cdot (\dfrac{e^r - 1}{e^r + 1})}{1 - (\dfrac{e^r - 1}{e^r + 1})^2} =} \\ {} & = & {2 \cdot \pi \cdot \dfrac{e^r - e^{-r}}{2} =} \\ {} & = & {2 \cdot \pi \cdot \sinh r.} \end{array}\)
Let \(\circum_h(r)\) denote the circumference of the h-circle of the h-radius \(r\) . Show that
\(\\text{circum}_h(r+1)>2\cdot \text{circum}_h(r)\])
for all \(r>0\) .
- Hint
-
Apply Proposition \(\PageIndex{1}\). Use that \(e > 2\) and in particular the function \(r \mapsto e^{-r}\) is decreasing.


4.2: Conjugates and Magnitudes
For each complex number \(z = x + iy\) , its complex conjugate is a complex number whose imaginary part has the sign flipped: \[z^* = x - i y.\] Conjugation obeys two important properties: \[\begin{align} (z_1 + z_2)^* &= z_1^* + z_2^* \\ (z_1 z_2)^* &= z_1^* z_2^*.\end{align}\]
Example \(\PageIndex{1}\)
Let us prove that \((z_1 z_2)^* = z_1^* z_2^*\) . First, let \(z_1 = x_1 + i y_1\) and \(z_2 = x_2 + i y_2\) . Then, \[\begin{align} (z_1 z_2)^* &= \left[(x_1+iy_1)(x_2+iy_2)\right]^* \\ &= \left[\left(x_1 x_2 - y_1 y_2\right) + i\left(x_1y_2+y_1x_2\right)\right]^* \\ &= \left(x_1 x_2 - y_1 y_2\right) - i\left(x_1y_2+y_1x_2\right) \\ &= \left(x_1 - i y_1\right)\left(x_2 - i y_2\right) \\ &= z_1^* z_2^* \end{align}\]
For a complex number \(z = x + i y\) , the magnitude of the complex number is \[|z| = \sqrt{x^2 + y^2}.\] This is a non-negative real number. A complex number and its conjugate have the same magnitude: \(|z| = |z^*|\) . Also, we can show that complex magnitudes have the property \[|z_1 z_2| = |z_1| \, |z_2|.\] This property is similar to the “absolute value” operation for real numbers, hence the similar notation.
As a corollary, taking a power of a complex number raises its magnitude by the same power: \[|z^n| = |z|^n \;\;\;\textrm{for}\;\;n \in \mathbb{Z}.\]