5.E: Exercises
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Exercise \(\PageIndex{1}\)
Suppose you have \(\mathbb{R}^2\) and the \(+\) operation is as follows: \[(a,b) + (c,d) = (a+d,b+c).\nonumber\] Scalar multiplication is defined in the usual way. Is this a vector space? Explain why or why not.
Exercise \(\PageIndex{2}\)
Suppose you have \(\mathbb{R}^2\) and the \(+\) operation is as follows: \[(a,b) + (c,d) = (0,b+d)\nonumber\] Scalar multiplication is defined in the usual way. Is this a vector space? Explain why or why not.
Exercise \(\PageIndex{3}\)
Suppose you have \(\mathbb{R}^2\) and scalar multiplication is defined as \(c(a,b) = (a, cb)\) while vector addition is defined as usual. Is this a vector space? Explain why or why not.
Exercise \(\PageIndex{4}\)
Suppose you have \(\mathbb{R}^2\) and the \(+\) operation is defined as follows. \[(a,b) + (c,d) = (a−c,b−d)\nonumber\] Scalar multiplication is same as usual. Is this a vector space? Explain why or why not.
Exercise \(\PageIndex{5}\)
Consider all the functions defined on a non empty set which have values in \(\mathbb{R}\). Is this a vector space? Explain. The operations are defined as follows. Here \(f ,g\) signify functions and \(a\) is a scalar \[\begin{aligned} (f+g)(x)&=f(x)+g(x) \\ (af)(x)&=a(f(x))\end{aligned}\]
Exercise \(\PageIndex{6}\)
Denote by \(\mathbb{R}^{\mathbb{N}}\) the set of real valued sequences. For \(\vec{a} ≡ \{a_n\}_{n=1}^∞\), \(\vec{b} ≡ \{b_n\}_{n=1}^\infty\) two of these, define their sum to be given by \[\vec{a}+\vec{b}=\{a_n+b_n\}_{n=1}^\infty\nonumber\] and define scalar multiplication by \[c\vec{a}=\{ca_n\}_{n=1}^\infty\text{ where }\vec{a}=\{a+n\}_{n=1}^\infty\nonumber\] Is this a special case of Exercise \(\PageIndex{5}\) ? Is this a vector space?
Exercise \(\PageIndex{7}\)
Let \(\mathbb{C}^2\) be the set of ordered pairs of complex numbers. Define addition and scalar multiplication in the usual way. \[(z,w) + (\hat{z},\hat{w}) = (z+\hat{z},w+\hat{w}), u(z,w) ≡ (uz,uw)\nonumber\] Here the scalars are from \(\mathbb{C}\). Show this is a vector space.
Exercise \(\PageIndex{8}\)
Let \(V\) be the set of functions defined on a nonempty set which have values in a vector space \(W\). Is this a vector space? Explain.
Exercise \(\PageIndex{9}\)
Consider the space of \(m\times n\) matrices with operation of addition and scalar multiplication defined the usual way. That is, if \(A,B\) are two \(m\times n\) matrices and \(c\) a scalar, \[(A+B)_{ i j} = A_{i j} +B_{i j}, \:(cA)_{ i j} ≡ c (A_{ij})\nonumber\]
Exercise \(\PageIndex{10}\)
Consider the set of \(n\times n\) symmetric matrices. That is, \(A = A^T\). In other words, \(A_{i j} = A_{ji}\). Show that this set of symmetric matrices is a vector space and a subspace of the vector space of \(n\times n\) matrices.
Exercise \(\PageIndex{11}\)
Consider the set of all vectors in \(\mathbb{R}^2 ,(x, y)\) such that \(x + y ≥ 0\). Let the vector space operations be the usual ones. Is this a vector space? Is it a subspace of \(\mathbb{R}^2\)?
Exercise \(\PageIndex{12}\)
Consider the vectors in \(\mathbb{R}^2 ,(x, y)\) such that \(xy = 0\). Is this a subspace of \(\mathbb{R}^2\)? Is it a vector space? The addition and scalar multiplication are the usual operations.
Exercise \(\PageIndex{13}\)
Define the operation of vector addition on \(\mathbb{R}^2\) by \((x, y) + (u, v) = (x+u, y+v+1)\). Let scalar multiplication be the usual operation. Is this a vector space with these operations? Explain.
Exercise \(\PageIndex{14}\)
Let the vectors be real numbers. Define vector space operations in the usual way. That is \(x+y\) means to add the two numbers and \(xy\) means to multiply them. Is \(\mathbb{R}\) with these operations a vector space? Explain.
Exercise \(\PageIndex{15}\)
Let the scalars be the rational numbers and let the vectors be real numbers which are the form \(a+b\sqrt{2}\) for \(a,b\) rational numbers. Show that with the usual operations, this is a vector space.
Exercise \(\PageIndex{16}\)
Let \(\mathbb{P}_2\) be the set of all polynomials of degree \(2\) or less. That is, these are of the form \(a+bx+cx^2\). Addition is defined as \[(a+bx+cx^2)+(\hat{d}+\hat{b}x+\hat{c}x^2)=(a+\hat{a})+(b+\hat{b})x+(c+\hat{c})x^2\nonumber\] and scalar multiplication is defined as \[d(a+bx+cx^2)=da+dbx+cdx^2\nonumber\] Show that, with this definition of the vector space operations that \(\mathbb{P}_2\) is a vector space. Now let \(V\) denote those polynomials \(a+bx+cx^2\) such that \(a+b+c = 0\). Is \(V\) a subspace of \(\mathbb{P}_2\)? Explain.
Exercise \(\PageIndex{17}\)
Let \(M,N\) be subspaces of a vector space \(V\) and consider \(M +N\) defined as the set of all \(m+n\) where \(m ∈ M\) and \(n ∈ N\). Show that \(M +N\) is a subspace of \(V\).
Exercise \(\PageIndex{18}\)
Let \(M,N\) be subspaces of a vector space \(V\). Then \(M ∩N\) consists of all vectors which are in both \(M\) and \(N\). Show that \(M ∩N\) is a subspace of \(V\).
Exercise \(\PageIndex{19}\)
Let \(M,N\) be subspaces of a vector space \(\mathbb{R}^2\). Then \(N ∪M\) consists of all vectors which are in either \(M\) or \(N\). Show that \(N ∪M\) is not necessarily a subspace of \(\mathbb{R}^2\) by giving an example where \(N ∪M\) fails to be a subspace.
Exercise \(\PageIndex{20}\)
Let \(X\) consist of the real valued functions which are defined on an interval \([a,b]\). For \(f ,g ∈ X, f +g\) is the name of the function which satisfies \((f +g) (x) = f (x) +g(x)\). For \(s\) a real number, \((s f) (x) = s(f (x))\). Show this is a vector space.
- Answer
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The axioms of a vector space all hold because they hold for a vector space. The only thing left to verify is the assertions about the things which are supposed to exist. \(0\) would be the zero function which sends everything to \(0\). This is an additive identity. Now if \(f\) is a function, \(−f (x) ≡ (−f (x))\). Then \[(f + (−f)) (x) ≡ f (x) + (−f) (x) ≡ f (x) + (−f (x)) = 0\nonumber\] Hence \(f + −f = 0\). For each \(x ∈ [a,b]\), let \(f_x (x) = 1\) and \(f_x (y) = 0\) if \(y\neq x\). Then these vectors are obviously linearly independent.
Exercise \(\PageIndex{21}\)
Consider functions defined on \(\{1, 2,\cdots ,n\}\) having values in \(\mathbb{R}\). Explain how, if \(V\) is the set of all such functions, \(V\) can be considered as \(\mathbb{R}^n\).
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Let \(f (i)\) be the \(i\)th component of a vector \(\vec{x} ∈ \mathbb{R}^n\). Thus a typical element in \(\mathbb{R}^n\) is \((f (1),\cdots , f (n))\).
Exercise \(\PageIndex{22}\)
Let the vectors be polynomials of degree no more than \(3\). Show that with the usual definitions of scalar multiplication and addition wherein, for \(p(x)\) a polynomial, \((ap) (x) = ap(x)\) and for \(p,q\) polynomials \((p+q) (x) = p(x) +q(x)\), this is a vector space.
- Answer
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This is just a subspace of the vector space of functions because it is closed with respect to vector addition and scalar multiplication. Hence this is a vector space.
Exercise \(\PageIndex{23}\)
Let \(V\) be a vector space and suppose \(\{\vec{x}_1,\cdots ,\vec{x}_l\}\) is a set of vectors in \(V\). Show that \(\vec{0}\) is in \(span\{\vec{x}_1,\cdots ,\vec{x}_k\}\).
- Answer
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\(\sum\limits_{i=1}^k0\vec{x}_k=\vec{0}\)
Exercise \(\PageIndex{24}\)
Determine if \(p(x) = 4x^2 −x\) is in the span given by \[span \{x^2+x,\:x^2-1,\:-x+2\}\nonumber\]
Exercise \(\PageIndex{25}\)
Determine if \(p(x) = −x^2 +x+2\) is in the span given by \[span\{ x^2 +x+1,\: 2x^2 +x\}\nonumber\]
Exercise \(\PageIndex{26}\)
Determine if \(A=\left[\begin{array}{cc}1&3\\0&0\end{array}\right]\) is in the span given by \[span\left\{\left[\begin{array}{cc}1&0\\0&1\end{array}\right],\:\left[\begin{array}{cc}0&1\\1&0\end{array}\right],\:\left[\begin{array}{cc}1&0\\1&1\end{array}\right],\:\left[\begin{array}{cc}0&1\\1&1\end{array}\right]\right\}\nonumber\]
Exercise \(\PageIndex{27}\)
Show that the spanning set in Exercise \(\PageIndex{26}\) is a spanning set for \(M_{22}\), the vector space of all \(2\times 2\) matrices.
Exercise \(\PageIndex{28}\)
Consider the vector space of polynomials of degree at most \(2\), \(\mathbb{P}_2\). Determine whether the following is a basis for \(\mathbb{P}_2\). \[\{x^2 +x+1,\: 2x^2 +2x+1,\: x+1\}\nonumber\] Hint: There is a isomorphism from \(\mathbb{R}^3\) to \(\mathbb{P}_2\). It is defined as follows: \[T\vec{e}_1 = 1,\: T\vec{e}_2 = x,\: T\vec{e}_3= x^2\nonumber\] Then extend \(T\) linearly. Thus \[T\left[\begin{array}{c}1\\1\\1\end{array}\right]=x^2+x+1,\:T\left[\begin{array}{c}1\\2\\2\end{array}\right]=2x^2+2x+1,\:T\left[\begin{array}{c}1\\1\\0\end{array}\right]=1+x\nonumber\] It follows that if \[\left\{\left[\begin{array}{c}1\\1\\1\end{array}\right],\:\left[\begin{array}{c}1\\2\\2\end{array}\right],\:\left[\begin{array}{c}1\\1\\0\end{array}\right]\right\}\nonumber\] is a basis for \(\mathbb{R}^3\), then the polynomials will be a basis for \(\mathbb{P}_2\) because they will be independent. Recall that an isomorphism takes a linearly independent set to a linearly independent set. Also, since \(T\) is an isomorphism, it preserves all linear relations.
Exercise \(\PageIndex{29}\)
Find a basis in \(\mathbb{P}_2\) for the subspace \[span\{ 1+x+x^2 ,\: 1+2x,\: 1+5x−3x^2\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others. Hint: This is the situation in which you have a spanning set and you want to cut it down to form a linearly independent set which is also a spanning set. Use the same isomorphism above. Since \(T\) is an isomorphism, it preserves all linear relations so if such can be found in \(\mathbb{R}^3\), the same linear relations will be present in \(\mathbb{P}_2\).
Exercise \(\PageIndex{30}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ 1+x−x^2 +x^3 ,\: 1+2x+3x^3 ,\:−1+3x+5x^2 +7x^3 ,\: 1+6x+4x^2 +11x^3\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{31}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ 1+x−x^2 +x^3 ,\: 1+2x+3x^3 ,\:−1+3x+5x^2 +7x^3 ,\: 1+6x+4x^2 +11x^3\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{32}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ x^3 −2x^2 +x+2,\: 3x^3 −x^2 +2x+2,\: 7x^3 +x^2 +4x+2,\: 5x^3 +3x+2\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{33}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ x^3 +2x^2 +x−2,\: 3x^3 +3x^2 +2x−2,\: 3x^3 +x+2,\: 3x^3 +x+2\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{34}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ x^3 −5x^2 +x+5,\: 3x^3 −4x^2 +2x+5,\: 5x^3 +8x^2 +2x−5,\: 11x^3 +6x+5\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{35}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{x^3 −3x^2 +x+3,\: 3x^3 −2x^2 +2x+3,\: 7x^3 +7x^2 +3x−3,\: 7x^3 +4x+3\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{36}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ x^3 −x^2 +x+1,\: 3x^3 +2x+1,\: 4x^3 +x^2 +2x+1,\: 3x^3 +2x−1\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{37}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ x^3 −x^2 +x+1,\: 3x^3 +2x+1,\: 13x^3 +x^2 +8x+4,\: 3x^3 +2x−1\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{38}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ x^3 −3x^2 +x+3,\: 3x^3 −2x^2 +2x+3,\:−5x^3 +5x^2 −4x−6,\: 7x^3 +4x−3\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{39}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ x^3 −2x^2 +x+2,\: 3x^3 −x^2 +2x+2,\: 7x^3 −x^2 +4x+4,\: 5x^3 +3x−2\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{40}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ x^3 −2x^2 +x+2,\: 3x^3 −x^2 +2x+2,\: 3x^3 +4x^2 +x−2,\: 7x^3 −x^2 +4x+4\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{41}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ x^3 −4x^2 +x+4,\: 3x^3 −3x^2 +2x+4,\:−3x^3 +3x^2 −2x−4,\:−2x^3 +4x^2 −2x−4\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{42}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ x^3 +2x^2 +x−2,\: 3x^3 +3x^2 +2x−2,\: 5x^3 +x^2 +2x+2,\: 10x^3 +10x^2 +6x−6\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{43}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ x^3 +x^2 +x−1,\: 3x^3 +2x^2 +2x−1,\: x^3 +1,\: 4x^3 +3x^2 +2x−1\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{44}\)
Find a basis in \(\mathbb{P}_3\) for the subspace \[span\{ x^3 −x^2 +x+1,\: 3x^3 +2x+1,\: x^3 +2x^2 −1,\: 4x^3 +x^2 +2x+1\}\nonumber\] If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.
Exercise \(\PageIndex{45}\)
Here are some vectors. \[\{ x^3 +x^2 −x−1,\: 3x^3 +2x^2 +2x−1\}\nonumber\] If these are linearly independent, extend to a basis for all of \(\mathbb{P}_3\).
Exercise \(\PageIndex{46}\)
Here are some vectors. \[\{ x^3 −2x^2 −x+2,\: 3x^3 −x^2 +2x+2\}\nonumber\] If these are linearly independent, extend to a basis for all of \(\mathbb{P}_3\).
Exercise \(\PageIndex{47}\)
Here are some vectors. \[\{ x^3 −3x^2 −x+3,\: 3x^3 −2x^2 +2x+3\}\nonumber\] If these are linearly independent, extend to a basis for all of \(\mathbb{P}_3\).
Exercise \(\PageIndex{48}\)
Here are some vectors. \[\{ x^3 −2x^2 −3x+2,\: 3x^3 −x^2 −6x+2,\:−8x^3 +18x+10\}\nonumber\] If these are linearly independent, extend to a basis for all of \(\mathbb{P}_3\).
Exercise \(\PageIndex{49}\)
Here are some vectors. \[\{ x^3 −3x^2 −3x+3,\: 3x^3 −2x^2 −6x+3,\:−8x^3 +18x+40\}\nonumber\] If these are linearly independent, extend to a basis for all of \(\mathbb{P}_3\).
Exercise \(\PageIndex{50}\)
Here are some vectors. \[\{ x^3 −x^2 +x+1,\: 3x^3 +2x+1,\: 4x^3 +2x+2\}\nonumber\] If these are linearly independent, extend to a basis for all of \(\mathbb{P}_3\).
Exercise \(\PageIndex{51}\)
Here are some vectors. \[\{ x^3 +x^2 +2x−1,\: 3x^3 +2x^2 +4x−1,\: 7x^3 +8x+23\}\nonumber\] If these are linearly independent, extend to a basis for all of \(\mathbb{P}_3\).
Exercise \(\PageIndex{52}\)
Determine if the following set is linearly independent. If it is linearly dependent, write one vector as a linear combination of the other vectors in the set. \[\{ x+1,\: x^2 +2,\: x^2 −x−3\}\nonumber\]
Exercise \(\PageIndex{53}\)
Determine if the following set is linearly independent. If it is linearly dependent, write one vector as a linear combination of the other vectors in the set. \[\{ x^2 +x,\:−2x^2 −4x−6,\: 2x−2\}\nonumber\]
Exercise \(\PageIndex{54}\)
Determine if the following set is linearly independent. If it is linearly dependent, write one vector as a linear combination of the other vectors in the set. \[\left\{\left[\begin{array}{cc}1&2\\0&1\end{array}\right],\:\left[\begin{array}{rr}-7&2\\-2&-3\end{array}\right],\:\left[\begin{array}{cc}4&0\\1&2\end{array}\right]\right\}\nonumber\]
Exercise \(\PageIndex{55}\)
Determine if the following set is linearly independent. If it is linearly dependent, write one vector as a linear combination of the other vectors in the set. \[\left\{\left[\begin{array}{cc}1&0\\0&1\end{array}\right],\:\left[\begin{array}{cc}0&1\\0&1\end{array}\right],\:\left[\begin{array}{cc}1&0\\1&0\end{array}\right],\:\left[\begin{array}{cc}0&0\\1&1\end{array}\right]\right\}\nonumber\]
Exercise \(\PageIndex{56}\)
If you have \(5\) vectors in \(\mathbb{R}^5\) and the vectors are linearly independent, can it always be concluded they span \(\mathbb{R}^5\)?
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Yes. If not, there would exist a vector not in the span. But then you could add in this vector and obtain a linearly independent set of vectors with more vectors than a basis.
Exercise \(\PageIndex{57}\)
If you have \(6\) vectors in \(\mathbb{R}^5\), is it possible they are linearly independent? Explain.
- Answer
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No. They can't be.
Exercise \(\PageIndex{58}\)
Let \(\mathbb{P}_3\) be the polynomials of degree no more than \(3\). Determine which of the following are bases for this vector space.
- \(\{ x+1,\: x^3 +x^2 +2x,\: x^2 +x,\: x^3 +x^2 +x\}\)
- \(\{ x^3 +1,\: x^2 +x,\: 2x^3 +x^2 ,\: 2x^3 −x^2 −3x+1\}\)
- Answer
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- Suppose \[c_1(x^3 +1)+c_2 (x^2 +x) +c_3( 2x^3 +x^2) +c_4 (2x^3 −x^2 −3x+1) = 0\nonumber\] Then combine the terms according to power of \(x\). \[(c_1 +2c_3 +2c_4) x^3 + (c_2 +c_3 −c_4) x^2 + (c_2 −3c_4) x+ (c_1 +c_4) = 0\nonumber\] Is there a non zero solution to the system \[\begin{aligned}c_1 +2c_3 +2c_4 &= 0\\ c_2 +c_3 −c_4 &= 0\\ c_2 −3c_4 &= 0\\ c_1 +c_4 &= 0\end{aligned}\], Solution is: \[[c_1 = 0,\: c_2 = 0,\: c_3 = 0,\: c_4 = 0]\nonumber\] Therefore, these are linearly independent.
Exercise \(\PageIndex{59}\)
In the context of the above problem, consider polynomials \[\{a_ix^3 +b_ix^2 +c_ix+d_i ,\: i = 1, 2, 3, 4\}\nonumber\] Show that this collection of polynomials is linearly independent on an interval \([s,t]\) if and only if \[\left[\begin{array}{cccc}a_1&b_1&c_1&d_1 \\ a_2&b_2&c_2&d_2 \\ a_3&b_3&c_3&d_3\\ a_4&b_4&c_4&d_4\end{array}\right]\nonumber\] is an invertible matrix.
- Answer
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Let \(p_i(x)\) denote the \(i\)th of these polynomials. Suppose \(\sum_i C_ip_i(x) = 0\). Then collecting terms according to the exponent of \(x\), you need to have \[\begin{aligned}C_1a_1 +C_2a_2 +C_3a_3 +C_4a_4 &= 0\\ C_1b_1 +C_2b_2 +C_3b_3 +C_4b_4 &= 0\\ C_1c_1 +C_2c_2 +C_3c_3 +C_4c_4 &= 0\\ C_1d_1 +C_2d_2 +C_3d_3 +C_4d_4 &= 0\end{aligned}\] The matrix of coefficients is just the transpose of the above matrix. There exists a non trivial solution if and only if the determinant of this matrix equals \(0\).
Exercise \(\PageIndex{60}\)
Let the field of scalars be \(\mathbb{Q}\), the rational numbers and let the vectors be of the form \(a+b\sqrt{2}\) where \(a,b\) are rational numbers. Show that this collection of vectors is a vector space with field of scalars \(\mathbb{Q}\) and give a basis for this vector space.
- Answer
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When you add two of these you get one and when you multiply one of these by a scalar, you get another one. A basis is \(\{1,\sqrt{2}\}\). By definition, the span of these gives the collection of vectors. Are they independent? Say \(a + b\sqrt{2} = 0\) where \(a,b\) are rational numbers. If \(a\neq 0\), then \(b\sqrt{2} = −a\) which can’t happen since a is rational. If \(b\neq 0\), then \(−a = b\sqrt{2}\) which again can’t happen because on the left is a rational number and on the right is an irrational. Hence both \(a,b = 0\) and so this is a basis.
Exercise \(\PageIndex{61}\)
Suppose \(V\) is a finite dimensional vector space. Based on the exchange theorem above, it was shown that any two bases have the same number of vectors in them. Give a different proof of this fact using the earlier material in the book. Hint: Suppose \(\{\vec{x}_1,\cdots ,\vec{x}_n\}\) and \(\{\vec{y}_1,\cdots , \vec{y}_m\}\) are two bases with \(m < n\). Then define \[φ : \mathbb{R}^n \mapsto V,\: ψ :\mathbb{R}^m\mapsto V\nonumber\] by \[φ (\vec{a}) = \sum\limits_{k=1}^n a_k\vec{x}_k ,\: ψ(\vec{b}) =\sum\limits_{j=1}^m b_j\vec{y}_j\nonumber\] Consider the linear transformation, \(ψ^{−1}\circ φ\). Argue it is a one to one and onto mapping from \(\mathbb{R}^n\) to \(\mathbb{R}^m\). Now consider a matrix of this linear transformation and its reduced row-echelon form.
- Answer
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This is obvious because when you add two of these you get one and when you multiply one of these by a scalar, you get another one. A basis is \(\{1,\sqrt{2}\}\). By definition, the span of these gives the collection of vectors. Are they independent? Say \(a+b\sqrt{2} = 0\) where \(a,b\) are rational numbers. If \(a\neq 0\), then \(b\sqrt{2} = −a\) which can’t happen since \(a\) is rational. If \(b\neq 0\), then \(−a = b\sqrt{2}\) which again can’t happen because on the left is a rational number and on the right is an irrational. Hence both \(a,b = 0\) and so this is a basis.
Exercise \(\PageIndex{62}\)
Let \(M =\{\vec{u} = (u_1,\:u_2,\:u_3,\:u_4)\in \mathbb{R}^4\: :\: |u_1| ≤ 4\}\). Is \(M\) a subspace of \(\mathbb{R}^4\)?
- Answer
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This is not a subspace. \(\left[\begin{array}{c}1\\1\\1\\1\end{array}\right]\) is in it, but \(20\left[\begin{array}{c}1\\1\\1\\1\end{array}\right]\) is not.
Exercise \(\PageIndex{63}\)
Let \(M =\{\vec{u} = (u_1,\:u_2,\:u_3,\:u_4)\in \mathbb{R}^4\: :\: \sin(u_1) = 1\}\). Is \(M\) a subspace of \(\mathbb{R}^4\)?
- Answer
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This is not a subspace.
Exercise \(\PageIndex{64}\)
Let \(W\) be a subset of \(M_{22}\) given by \[W = \{ A|A\in M_{22},A^T = A\}\nonumber\] In words, \(W\) is the set of all symmetric \(2\times 2\) matrices. Is \(W\) a subspace of \(M_{22}\)?
Exercise \(\PageIndex{65}\)
Let \(W\) be a subset of \(M_{22}\) given by \[W=\left\{\left[\begin{array}{cc}a&b\\c&d\end{array}\right] \: |a,b,c,d\in\mathbb{R},\:a+b=c+d\right\}\nonumber\] Is \(W\) a subspace of \(M_{22}\)?
Exercise \(\PageIndex{66}\)
Let \(W\) be a subset of \(P_3\) given by \[W = \{ ax^3 +bx^2 +cx+d|\: a,b, c,d\in\mathbb{R},d = 0\}\nonumber\] Is \(W\) a subspace of \(P_3\)?
Exercise \(\PageIndex{67}\)
Let \(W\) be a subset of \(P_3\) given by \[W = \{ p(x) = ax^3 +bx^2 +cx+d|\: a,b, c,d\in\mathbb{R}, p(2) = 1\}\nonumber\] Is \(W\) a subspace of \(P_3\)?
Exercise \(\PageIndex{68}\)
Let \(T\): \(\mathbb{P}_2\to\mathbb{R}\) be a linear transformation such that \[T(x^2)=1;\: T(x^2+x)=5;\: T(x^2+x+1)=-1.\nonumber\] Find \(T(ax^2+bx+c)\).
- Answer
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By linearity we have \(T(x^2 ) = 1,\: T(x) = T(x^2 +x−x^2 ) = T(x^2 +x)−T (x^2 ) = 5−1 = 5,\) and \(T(1) = T(x^2 +x+1−(x^2 +x)) = T(x^2 +x+1)−T(x^2 +x)) = −1−5 = −6\). Thus \(T(ax^2 +bx+c) = aT(x^2 ) +bT(x) +cT(1) = a+5b−6c\).
Exercise \(\PageIndex{69}\)
Consider the following functions \(T\): \(\mathbb{R}^3\to\mathbb{R}^2\). Explain why each of these functions \(T\) is not linear.
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y+3z+1 \\ 2y-3x+z\end{array}\right]\)
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y^2+3z \\ 2y+3z+z\end{array}\right]\)
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}\sin x+2y+3z \\ 2y+3z+z\end{array}\right]\)
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y+3z \\ 2y+3z-\ln z\end{array}\right]\)
Exercise \(\PageIndex{70}\)
Suppose \(T\) is a linear transformation such that \[\begin{aligned} T\left[\begin{array}{r}1\\1\\-7\end{array}\right]&=\left[\begin{array}{c}3\\3\\3\end{array}\right] \\ T\left[\begin{array}{r}-1\\0\\6\end{array}\right]&=\left[\begin{array}{c}1\\2\\3\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\2\end{array}\right]&=\left[\begin{array}{r}1\\3\\-1\end{array}\right]\end{aligned}\] Find the matrix of \(T\). That is find \(A\) such that \(T(\vec{x})=A\vec{x}\).
- Answer
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\[\left[\begin{array}{rrr}3&1&1\\3&2&3\\3&3&-1\end{array}\right]\left[\begin{array}{ccc}6&2&1\\5&2&1\\6&1&1\end{array}\right]=\left[\begin{array}{ccc}29&9&5\\46&13&8\\27&11&5\end{array}\right]\nonumber\]
Exercise \(\PageIndex{71}\)
Suppose \(T\) is a linear transformation such that \[\begin{aligned} T\left[\begin{array}{r}1\\2\\-18\end{array}\right]&=\left[\begin{array}{c}5\\2\\5\end{array}\right] \\ T\left[\begin{array}{r}-1\\-1\\15\end{array}\right]&=\left[\begin{array}{c}3\\3\\5\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\4\end{array}\right]&=\left[\begin{array}{r}2\\5\\-2\end{array}\right]\end{aligned}\] Find the matrix of \(T\). That is find \(A\) such that \(T(\vec{x})=A\vec{x}\).
- Answer
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\[\left[\begin{array}{rrr}5&3&2\\2&3&5\\5&5&-2\end{array}\right]\left[\begin{array}{ccc}11&4&1\\10&4&1\\12&3&1\end{array}\right]=\left[\begin{array}{ccc}109&38&10\\112&35&10\\81&34&8\end{array}\right]\nonumber\]
Exercise \(\PageIndex{72}\)
Consider the following functions \(T\): \(\mathbb{R}^3\to\mathbb{R}^2\). Show that each is a linear transformation and determine for each the matrix \(A\) such that \(T(\vec{x}) = A\vec{x}\).
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y+3z \\ 2y-3x+z\end{array}\right]\)
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}7x+2y+z \\ 3x-11y+2z\end{array}\right]\)
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}3x+2y+z \\ x+2y+6z\end{array}\right]\)
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}2y-5x+z \\ x+y+z\end{array}\right]\)
Exercise \(\PageIndex{73}\)
Suppose \[[A_1\cdots A_n]^{-1}\nonumber\] exists where each \(A_j\in\mathbb{R}^n\) and let vectors \(\{B_1,\cdots ,B_n\}\) in \(\mathbb{R}^m\) be given. Show that there always exists a linear transformation \(T\) such that \(T(A_i)=B_i\).
Exercise \(\PageIndex{74}\)
Let \(V\) and \(W\) be subspaces of \(\mathbb{R}^n\) and \(\mathbb{R}^m\) respectively and let \(T\): \(V → W\) be a linear transformation. Suppose that \(\{T\vec{v}_1,\cdots ,T\vec{v}_r\}\) is linearly independent. Show that it must be the case that \(\{\vec{v}_1,\cdots ,\vec{v}_r\}\) is also linearly independent.
- Answer
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If \(\sum\limits_i^ra_i\vec{v}_r=0\), then using linearity properties of \(T\) we get \[0=T(0)=T\left(\sum\limits_i^ra_i\vec{v}_r\right)=\sum\limits_i^ra_iT(\vec{v}_r).\nonumber\] Since we assume that \(\{T\vec{v}_1,\cdots ,T\vec{v}_r\}\) is linearly independent, we must have all \(a_i = 0\), and therefore we conclude that \(\{\vec{v}_1,\cdots ,\vec{v}_r\}\) is also linearly independent.
Exercise \(\PageIndex{75}\)
Let \[V=span\left\{\left[\begin{array}{c}1\\1\\2\\0\end{array}\right],\:\left[\begin{array}{c}0\\1\\1\\1\end{array}\right],\:\left[\begin{array}{c}1\\1\\0\\1\end{array}\right]\right\}\nonumber\] Let \(T\vec{x}=A\vec{x}\) where \(A\) is the matrix \[\left[\begin{array}{cccc}1&1&1&1\\0&1&1&0\\0&1&2&1\\1&1&1&2\end{array}\right]\nonumber\] Give a basis for \(Im(T)\).
Exercise \(\PageIndex{76}\)
Let \[V=span\left\{\left[\begin{array}{c}1\\0\\0\\1\end{array}\right],\:\left[\begin{array}{c}1\\1\\1\\1\end{array}\right],\:\left[\begin{array}{c}1\\4\\4\\1\end{array}\right]\right\}\nonumber\] Let \(T\vec{x}=A\vec{x}\) where \(A\) is the matrix \[\left[\begin{array}{cccc}1&1&1&1\\0&1&1&0\\0&1&2&1\\1&1&1&2\end{array}\right]\nonumber\] Find a basis for \(Im(T)\). In this case, the original vectors do not form an independent set.
- Answer
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Since the third vector is a linear combinations of the first two, then the image of the third vector will also be a linear combinations of the image of the first two. However the image of the first two vectors are linearly independent (check!), and hence form a basis of the image. Thus a basis for \(Im(T)\) is: \[V=span\left\{\left[\begin{array}{c}2\\0\\1\\3\end{array}\right],\:\left[\begin{array}{c}4\\2\\4\\5\end{array}\right]\right\}\nonumber\]
Exercise \(\PageIndex{77}\)
If \(\{\vec{v}_1,\cdots ,\vec{v}_r\}\) is linearly independent and \(T\) is a one to one linear transformation, show that \(\{T\vec{v}_1,\cdots ,T\vec{v}_r\}\) is also linearly independent. Give an example which shows that if \(T\) is only linear, it can happen that, although \(\{\vec{v}_1,\cdots ,\vec{v}_r\}\) is linearly independent, \(\{T\vec{v}_1,\cdots ,T\vec{v}_r\}\) is not. In fact, show that it can happen that each of the \(T\vec{v}_j\) equals \(0\).
Exercise \(\PageIndex{78}\)
Let \(V\) and \(W\) be subspaces of \(\mathbb{R}^n\) and \(\mathbb{R}^m\) respectively and let \(T\): \(V → W\) be a linear transformation. Show that if \(T\) is onto \(W\) and if \(\{\vec{v}_1,\cdots ,\vec{v}_r\}\) is a basis for \(V\), then \(span\{T\vec{v}_1,\cdots ,T\vec{v}_r\} = W\).
Exercise \(\PageIndex{79}\)
Define \(T\): \(\mathbb{R}^4 → \mathbb{R}^3\) as follows. \[T\vec{x}=\left[\begin{array}{rrrr}3&2&1&8\\2&2&-2&6\\1&1&-1&3\end{array}\right]\vec{x}\nonumber\] Find a basis for \(Im(T)\). Also find a basis for \(\text{ker}(T)\).
Exercise \(\PageIndex{80}\)
Define \(T\): \(\mathbb{R}^4 → \mathbb{R}^3\) as follows. \[T\vec{x}=\left[\begin{array}{rrr}1&2&0\\1&1&1\\0&1&1\end{array}\right]\vec{x}\nonumber\] where on the right, it is just matrix multiplication of the vector \(\vec{x}\) which is meant. Explain why \(T\) is an isomorphism of \(\mathbb{R}^3\) to \(\mathbb{R}^3\).
Exercise \(\PageIndex{81}\)
Suppose \(T\): \(\mathbb{R}^3 → \mathbb{R}^3\) is a linear transformation given by \[T\vec{x}=A\vec{x}\nonumber\] where \(A\) is a \(3\times 3\) matrix. Show that \(T\) is an isomorphism if and only if \(A\) is invertible.
Exercise \(\PageIndex{82}\)
Suppose \(T\): \(\mathbb{R}^3 → \mathbb{R}^3\) is a linear transformation given by \[T\vec{x}=A\vec{x}\nonumber\] where \(A\) is a \(m\times n\) matrix. Show that \(T\) is never an isomorphism if \(m\neq n\). In particular, show that if \(m>n\), \(T\) cannot be onto and if \(m<n\), then \(T\) cannot be one to one.
Exercise \(\PageIndex{83}\)
Define \(T\): \(\mathbb{R}^2 → \mathbb{R}^3\) as follows. \[T\vec{x}=\left[\begin{array}{cc}1&0\\1&1\\0&1\end{array}\right]\vec{x}\nonumber\] where on the right, it is just matrix multiplication of the vector \(\vec{x}\) which is meant. Show that \(T\) is one to one. Next let \(W = Im(T)\). Show that \(T\) is an isomorphism of \(\mathbb{R}^2\) and \(Im (T)\).
Exercise \(\PageIndex{84}\)
In the above problem, find a \(2\times 3\) matrix \(A\) such that the restriction of \(A\) to \(Im(T)\) gives the same result as \(T^{−1}\) on \(Im(T)\). Hint: You might let \(A\) be such that \[A\left[\begin{array}{c}1\\1\\0\end{array}\right]=\left[\begin{array}{c}1\\0\end{array}\right],\:A\left[\begin{array}{c}0\\1\\1\end{array}\right]=\left[\begin{array}{c}0\\1\end{array}\right]\nonumber\] now find another vector \(\vec{v} ∈ \mathbb{R}^3\) such that \[\left\{\left[\begin{array}{c}1\\1\\0\end{array}\right],\:\left[\begin{array}{c}0\\1\\1\end{array}\right],\:\vec{v}\right\}\nonumber\] is a basis. You could pick \[\vec{v}=\left[\begin{array}{c}0\\0\\1\end{array}\right]\nonumber\] for example. Explain why this one works or one of your choice works. Then you could define \(A\vec{v}\) to equal some vector in \(\mathbb{R}^2\). Explain why there will be more than one such matrix \(A\) which will deliver the inverse isomorphism \(T^{−1}\) on \(Im(T)\).
Exercise \(\PageIndex{85}\)
Now let \(V\) equal \(span\left\{\left[\begin{array}{c}1\\0\\1\end{array}\right],\:\left[\begin{array}{c}0\\1\\1\end{array}\right]\right\}\) and let \(T\): \(V\to W\) be a linear transformation where \[W=span\left\{\left[\begin{array}{c}1\\0\\1\\0\end{array}\right],\:\left[\begin{array}{c}0\\1\\1\\1\end{array}\right]\right\}\nonumber\] and \[T\left[\begin{array}{c}1\\0\\1\end{array}\right]=\left[\begin{array}{c}1\\0\\1\\0\end{array}\right],\:T\left[\begin{array}{c}0\\1\\1\end{array}\right]=\left[\begin{array}{c}0\\1\\1\\1\end{array}\right]\nonumber\]
Explain why \(T\) is an isomorphism. Determine a matrix \(A\) which, when multiplied on the left gives the same result as \(T\) on \(V\) and a matrix \(B\) which delivers \(T^{−1}\) on \(W\). Hint: You need to have \[A\left[\begin{array}{cc}1&0\\0&1\\1&1\end{array}\right]=\left[\begin{array}{cc}1&0\\0&1\\1&1\\0&1\end{array}\right]\nonumber\]
Now enlarge \(\left[\begin{array}{c}1\\0\\1\end{array}\right]\), \(\left[\begin{array}{c}0\\1\\1\end{array}\right]\) to obtain a basis for \(\mathbb{R}^3\). You could add in \(\left[\begin{array}{c}0\\0\\1\end{array}\right]\) for example, and then pick another vector in \(\mathbb{R}^4\) and let \(A\left[\begin{array}{c}0\\0\\1\end{array}\right]\) equal this other vector. Then you would have \[A\left[\begin{array}{ccc}1&0&0\\0&1&0\\1&1&1\end{array}\right]=\left[\begin{array}{ccc}1&0&0\\0&1&0\\1&1&0\\0&1&1\end{array}\right]\nonumber\]
This would involve picking for the new vector in \(\mathbb{R}^4\) the vector \(\left[\begin{array}{cccc}0&0&0&1\end{array}\right]^T\). Then you could find \(A\). You can do something similar to find a matrix for \(T^{-1}\) denoted as \(B\).
Exercise \(\PageIndex{86}\)
Let \(V=\mathbb{R}^3\) and let \[W=span(S),\text{ where }S=\left\{\left[\begin{array}{r}1\\-1\\1\end{array}\right],\:\left[\begin{array}{r}-2\\2\\-2\end{array}\right],\:\left[\begin{array}{r}-1\\1\\1\end{array}\right],\:\left[\begin{array}{r}1\\-1\\3\end{array}\right]\right\}\nonumber\] Find a basis of \(W\) consisting of vectors in \(S\).
- Answer
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In this case \(\text{dim}(W) = 1\) and a basis for \(W\) consisting of vectors in \(S\) can be obtained by taking any (nonzero) vector from \(S\).
Exercise \(\PageIndex{87}\)
Let \(T\) be a linear transformation given by \[T\left[\begin{array}{c}x\\y\end{array}\right]=\left[\begin{array}{cc}1&1\\1&1\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]\nonumber\] Find a basis for \(\text{ker}(T)\) and \(Im(T)\).
- Answer
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A basis for \(\text{ker}(T)\) is \(\left\{\left[\begin{array}{r}1\\-1\end{array}\right]\right\}\) and a basis for \(Im(T)\) is \(\left\{\left[\begin{array}{r}1\\1\end{array}\right]\right\}\). There are many other possibilities for the specific bases, but in this case \(\text{dim}(\text{ker}(T)) = 1\) and \(\text{dim}(Im(T)) = 1\).
Exercise \(\PageIndex{88}\)
Let \(T\) be a linear transformation given by \[T\left[\begin{array}{c}x\\y\end{array}\right]=\left[\begin{array}{cc}1&0\\1&1\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]\nonumber\] Find a basis for \(\text{ker}(T)\) and \(Im(T)\).
- Answer
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In this case \(\text{ker}(T) = \{0\}\) and \(Im(T) = \mathbb{R}^2\) (pick any basis of \(\mathbb{R}^2\)).
Exercise \(\PageIndex{89}\)
Let \(V=\mathbb{R}^3\) and let \[W=span\left\{\left[\begin{array}{c}1\\1\\1\end{array}\right],\:\left[\begin{array}{r}-1\\2\\-1\end{array}\right]\right\}\nonumber\] Extend this basis of \(W\) to a basis of \(V\).
- Answer
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There are many possible such extensions, one is (how do we know?): \[\left\{\left[\begin{array}{r}1\\1\\1\end{array}\right],\:\left[\begin{array}{r}-1\\2\\-1\end{array}\right],\:\left[\begin{array}{c}0\\0\\1\end{array}\right]\right\}\nonumber\]
Exercise \(\PageIndex{90}\)
Let \(T\) be a linear transformation given by \[T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{ccc}1&1&1\\1&1&1\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right]\nonumber\] What is \(\text{dim}(\text{ker}(T))\)?
- Answer
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We can easily see that \(\text{dim}(Im(T)) = 1\), and thus \(\text{dim}(\text{ker}(T)) = 3−\text{dim}(Im(T)) = 3−1 = 2\).
Exercise \(\PageIndex{91}\)
Consider the following functions which map \(\mathbb{R}^n\) to \(\mathbb{R}^n\).
- \(T\) multiplies the \(j\)th component of \(\vec{x}\) by a nonzero number \(b\).
- \(T\) replaces the \(i\)th component of \(\vec{x}\) with \(b\) times the \(j\)th component added to the \(i\)h component.
- \(T\) switches the \(i\)th and \(j\)th components.
Show these functions are linear transformations and describe their matrices \(A\) such that \(T (\vec{x}) = A\vec{x}\).
- Answer
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- The matrix of \(T\) is the elementary matrix which multiplies the \(j\)th diagonal entry of the identity matrix by \(b\).
- The matrix of \(T\) is the elementary matrix which takes \(b\) times the \(j\)th row and adds to the \(i\)th row.
- The matrix of \(T\) is the elementary matrix which switches the \(i\)th and the \(j\)th rows where the two components are in the \(i\)th and \(j\)th positions.
Exercise \(\PageIndex{92}\)
You are given a linear transformation \(T\): \(\mathbb{R}^n → \mathbb{R}^m\) and you know that \[T(A_i)=B_i\nonumber\] where \(\left[\begin{array}{ccc}A_1&\cdots&A_n\end{array}\right]^{-1}\) exists. Show that the matrix of \(T\) is of the form \[\left[\begin{array}{ccc}B_1&\cdots&B_n\end{array}\right]\:\left[\begin{array}{ccc}A_1&\cdots&A_n\end{array}\right]^{-1}\nonumber\]
- Answer
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Suppose \[\left[\begin{array}{c}\vec{c}_1^T \\ \vdots \\ \vec{c}_n^T\end{array}\right]=\left[\begin{array}{ccc}\vec{a}_1&\cdots&\vec{a}_n\end{array}\right]^{-1}\nonumber\] Thus \(\vec{c}_i^T\vec{a}_j=\delta_{ij}\). Therefore \[\begin{aligned} \left[\begin{array}{ccc}\vec{b}_1&\cdots&\vec{b}_n\end{array}\right]\: \left[\begin{array}{ccc}\vec{a}_1&\cdots&\vec{a}_n\end{array}\right]^{-1}\vec{a}_i &=\left[\begin{array}{ccc}\vec{b}_1&\cdots&\vec{b}_n\end{array}\right]\:\left[\begin{array}{c}\vec{c}_1^T \\ \vdots \\ \vec{c}_n^T\end{array}\right] \vec{a}_i \\ &=\left[\begin{array}{ccc}\vec{b}_1&\cdots&\vec{b}_n\end{array}\right] \vec{e}_i \\ &=\vec{b}_i\end{aligned}\] Thus \(T\vec{a}_i=\left[\begin{array}{ccc}\vec{b}_1&\cdots&\vec{b}_n\end{array}\right]\: \left[\begin{array}{ccc}\vec{a}_1&\cdots&\vec{a}_n\end{array}\right]^{-1}\vec{a}_i=A\vec{a}_i\). If \(\vec{x}\) is arbitrary, then since the matrix \(\left[\begin{array}{ccc}\vec{a}_1&\cdots&\vec{a}_n\end{array}\right]\) is invertible, there exists a unique \(\vec{y}\) such that \(\left[\begin{array}{ccc}\vec{a}_1&\cdots&\vec{a}_n\end{array}\right]\vec{y}=\vec{x}\) Hence \[T\vec{x}=T\left(\sum\limits_{i=1}^ny_i\vec{a}_i\right)=\sum\limits_{i=1}^ny_iT\vec{a}_i=\sum\limits_{i=1}^ny_1A\vec{a}_i=A\left(\sum\limits_{i=1}^ny_i\vec{a}_i\right)=A\vec{x}\nonumber\]
Exercise \(\PageIndex{93}\)
Suppose \(T\) is a linear transformation such that \[\begin{aligned}T\left[\begin{array}{r}1\\2\\-6\end{array}\right]&=\left[\begin{array}{c}5\\1\\3\end{array}\right] \\ T\left[\begin{array}{r}-1\\-1\\5\end{array}\right]&=\left[\begin{array}{c}1\\1\\5\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\2\end{array}\right]&=\left[\begin{array}{r}5\\3\\-2\end{array}\right]\end{aligned}\] Find the matrix of \(T\). That is find \(A\) such that \(T(\vec{x}) = A\vec{x}\).
- Answer
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\[\left[\begin{array}{rrr}5&1&5\\1&1&3\\3&5&-2\end{array}\right]\left[\begin{array}{ccc}3&2&1\\2&2&1\\4&1&1\end{array}\right]=\left[\begin{array}{ccc}37&17&11\\17&7&5\\11&14&6\end{array}\right]\nonumber\]
Exercise \(\PageIndex{94}\)
Suppose \(T\) is a linear transformation such that \[\begin{aligned}T\left[\begin{array}{r}1\\1\\-8\end{array}\right]&=\left[\begin{array}{c}1\\3\\1\end{array}\right] \\ T\left[\begin{array}{r}-1\\0\\6\end{array}\right]&=\left[\begin{array}{c}2\\4\\1\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\3\end{array}\right]&=\left[\begin{array}{r}6\\1\\-1\end{array}\right]\end{aligned}\] Find the matrix of \(T\). That is find \(A\) such that \(T(\vec{x}) = A\vec{x}\).
- Answer
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\[\left[\begin{array}{rrr}1&2&6\\3&4&1\\1&1&-1\end{array}\right]\left[\begin{array}{ccc}6&3&1\\5&3&1\\6&2&1\end{array}\right]=\left[\begin{array}{ccc}52&21&9\\44&23&8\\5&4&1\end{array}\right]\nonumber\]
Exercise \(\PageIndex{95}\)
Suppose \(T\) is a linear transformation such that \[\begin{aligned}T\left[\begin{array}{r}1\\3\\-7\end{array}\right]&=\left[\begin{array}{c}-3\\1\\3\end{array}\right] \\ T\left[\begin{array}{r}-1\\-2\\6\end{array}\right]&=\left[\begin{array}{4}1\\3\\-3\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\2\end{array}\right]&=\left[\begin{array}{r}5\\3\\-3\end{array}\right]\end{aligned}\] Find the matrix of \(T\). That is find \(A\) such that \(T(\vec{x}) = A\vec{x}\).
- Answer
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\[\left[\begin{array}{rrr}-3&1&5\\1&3&3\\3&-3&-3\end{array}\right]\left[\begin{array}{ccc}2&2&1\\1&2&1\\4&1&1\end{array}\right]=\left[\begin{array}{rrr}15&1&3\\17&11&7\\-9&-3&-3\end{array}\right]\nonumber\]
Exercise \(\PageIndex{96}\)
Suppose \(T\) is a linear transformation such that \[\begin{aligned}T\left[\begin{array}{r}1\\1\\-7\end{array}\right]&=\left[\begin{array}{c}3\\3\\3\end{array}\right] \\ T\left[\begin{array}{r}-1\\0\\6\end{array}\right]&=\left[\begin{array}{c}1\\2\\3\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\2\end{array}\right]&=\left[\begin{array}{r}1\\3\\-1\end{array}\right]\end{aligned}\] Find the matrix of \(T\). That is find \(A\) such that \(T(\vec{x}) = A\vec{x}\).
- Answer
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\[\left[\begin{array}{rrr}3&1&1\\3&2&3\\3&3&-1\end{array}\right]\left[\begin{array}{ccc}6&2&1\\5&2&1\\6&1&1\end{array}\right]=\left[\begin{array}{ccc}29&9&5\\46&13&8\\27&11&5\end{array}\right]\nonumber\]
Exercise \(\PageIndex{97}\)
Suppose \(T\) is a linear transformation such that \[\begin{aligned}T\left[\begin{array}{r}1\\2\\-18\end{array}\right]&=\left[\begin{array}{c}5\\2\\5\end{array}\right] \\ T\left[\begin{array}{r}-1\\-1\\15\end{array}\right]&=\left[\begin{array}{c}3\\3\\5\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\4\end{array}\right]&=\left[\begin{array}{r}2\\5\\-2\end{array}\right]\end{aligned}\] Find the matrix of \(T\). That is find \(A\) such that \(T(\vec{x}) = A\vec{x}\).
- Answer
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\[\left[\begin{array}{rrr}5&3&2\\2&3&5\\5&5&-2\end{array}\right]\left[\begin{array}{ccc}11&4&1\\10&4&1\\12&3&1\end{array}\right]=\left[\begin{array}{ccc}109&38&10 \\112&35&10\\81&34&8\end{array}\right]\nonumber\]
Exercise \(\PageIndex{98}\)
Consider the following functions \(T\): \(\mathbb{R}^3 → \mathbb{R}^2\). Show that each is a linear transformation and determine for each the matrix \(A\) such that \(T(\vec{x}) = A\vec{x}\).
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y+3z \\ 2y-3x+z\end{array}\right]\)
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}7x+2y+z \\ 3x-11y+2z\end{array}\right]\)
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}3x+2y+z \\ x+2y+6z\end{array}\right]\)
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}2y-5x+z \\ x+y+z\end{array}\right]\)
Exercise \(\PageIndex{99}\)
Consider the following functions \(T\): \(\mathbb{R}^3 → \mathbb{R}^2\). Explain why each of these functions \(T\) is not linear.
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y+3z+1 \\ 2y-3x+z\end{array}\right]\)
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y^2+3z \\ 2y+3x+z\end{array}\right]\)
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}\sin x+2y+3z \\ 2y+3x+z\end{array}\right]\)
- \(T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y+3z \\ 2y+3x-\ln z\end{array}\right]\)
Exercise \(\PageIndex{100}\)
Suppose \[\left[\begin{array}{ccc}A_1&\cdots&A_n\end{array}\right]^{-1}\nonumber\] exists where each \(A_j ∈ \mathbb{R}^n\) and let vectors \(\{B_1,\cdots ,B_n\}\) in \(\mathbb{R}^m\) be given. Show that there always exists a linear transformation \(T\) such that \(T(A_i) = B_i\).
Exercise \(\PageIndex{101}\)
Find the matrix for \(T (\vec{w}) = \text{proj}_{\vec{v}} (\vec{w})\) where \(\vec{v}=\left[\begin{array}{ccc}1&-2&3\end{array}\right]^T\).
- Answer
-
Recall that \(\text{proj}_{\vec{u}}(\vec{v}) = \frac{\vec{v}\bullet\vec{u}}{||\vec{u}||^2}\vec{u}\) and so the desired matrix has \(i\)th column equal to \(\text{proj}_{\vec{u}} (\vec{e}_i)\). Therefore, the matrix desired is \[\frac{1}{14}\left[\begin{array}{rrr}1&-2&3\\-2&4&-6\\3&-6&9\end{array}\right]\nonumber\]
Exercise \(\PageIndex{102}\)
Find the matrix for \(T (\vec{w}) = \text{proj}_{\vec{v}} (\vec{w})\) where \(\vec{v}=\left[\begin{array}{ccc}1&5&3\end{array}\right]^T\).
- Answer
-
\[\frac{1}{35}\left[\begin{array}{ccc}1&5&3\\5&25&15\\3&15&9\end{array}\right]\nonumber\]
Exercise \(\PageIndex{103}\)
Find the matrix for \(T (\vec{w}) = \text{proj}_{\vec{v}} (\vec{w})\) where \(\vec{v}=\left[\begin{array}{ccc}1&0&3\end{array}\right]^T\).
- Answer
-
\[\frac{1}{10}\left[\begin{array}{ccc}1&0&3\\0&0&0\\3&0&9\end{array}\right]\nonumber\]
Exercise \(\PageIndex{104}\)
Let \(B=\left\{\left[\begin{array}{r}2\\-1\end{array}\right],\:\left[\begin{array}{c}3\\2\end{array}\right]\right\}\) be a basis of \(\mathbb{R}^2\) and let \(\vec{x}=\left[\begin{array}{r}5\\-7\end{array}\right]\) be a vector in \(\mathbb{R}^2\). Find \(C_B(\vec{x})\).
Exercise \(\PageIndex{105}\)
Let \(B=\left\{\left[\begin{array}{r}1\\-1\\2\end{array}\right],\:\left[\begin{array}{c}2\\1\\2\end{array}\right],\:\left[\begin{array}{r}-1\\0\\2\end{array}\right]\right\}\) be a basis of \(\mathbb{R}^3\) and let \(\vec{x}=\left[\begin{array}{r}5\\-1\\4\end{array}\right]\) be a vector in \(\mathbb{R}^2\). Find \(C_B(\vec{x})\).
- Answer
-
\(C_B(\vec{x})=\left[\begin{array}{r}2\\1\\-1\end{array}\right]\).
Exercise \(\PageIndex{106}\)
Let \(T\): \(\mathbb{R}^2\mapsto \mathbb{R}^2\) be a linear transformation defined by \(T\left(\left[\begin{array}{c}a\\b\end{array}\right]\right)=\left[\begin{array}{c}a+b\\a-b\end{array}\right]\).
Consider the two bases \[B_1=\{\vec{v}_1,\vec{v}_2\}=\left\{\left[\begin{array}{c}1\\0\end{array}\right],\:\left[\begin{array}{r}-1\\1\end{array}\right]\right\}\nonumber\] and \[B_2=\left\{\left[\begin{array}{c}1\\1\end{array}\right],\:\left[\begin{array}{r}1\\-1\end{array}\right]\right\}\nonumber\] Find the matrix \(M_{B_2,B_1}\) of \(T\) with respect to the bases \(B_1\) and \(B_2\).
- Answer
-
\(M_{B_2B_1}=\left[\begin{array}{rr}1&0\\-1&1\end{array}\right]\)


11.3: Reaction Quotient
Make sure you thoroughly understand the following essential ideas:
- When arbitrary quantities of the different components of a chemical reaction system are combined, the overall system composition will not likely correspond to the equilibrium composition. As a result, a net change in composition ("a shift to the right or left") will tend to take place until the equilibrium state is attained.
- The status of the reaction system in regard to its equilibrium state is characterized by the value of the equilibrium expression whose formulation is defined by the coefficients in the balanced reaction equation; it may be expressed in terms of concentrations, or in the case of gaseous components, as partial pressures.
- The various terms in the equilibrium expression can have any arbitrary value (including zero); the value of the equilibrium expression itself is called the reaction quotient Q.
- If the concentration or pressure terms in the equilibrium expression correspond to the equilibrium state of the system, then Q has the special value K , which we call the equilibrium constant .
- The ratio of Q/K (whether it is 1, >1 or <1) thus serves as an index of how far the system is from its equilibrium composition, and its value indicates the direction in which the net reaction must proceed in order to reach its equilibrium state.
- When Q = K , then the equilibrium state has been reached, and no further net change in composition will take place as long as the system remains undisturbed.
Consider a simple reaction such as the gas-phase synthesis of hydrogen iodide from its elements: \[H_2 + I_2 \rightarrow 2 HI\] Suppose you combine arbitrary quantities of \(H_2\), \(I_2\) and \(HI\). Will the reaction create more HI, or will some of the HI be consumed as the system moves toward its equilibrium state? The concept of the reaction quotient , which is the focus of this short lesson, makes it easy to predict what will happen.
What is the Equilibrium Quotient?
In the previous section we defined the equilibrium expression for the reaction
In the general case in which the concentrations can have any arbitrary values (including zero), this expression is called the reaction quotient (the term equilibrium quotient is also commonly used.) and its value is denoted by \(Q\) (or \(Q_c\) or \(Q_p\) if we wish to emphasize that the terms represent molar concentrations or partial pressures.) If the terms correspond to equilibrium concentrations, then the above expression is called the equilibrium constant and its value is denoted by \(K\) (or \(K_c\) or \(K_p\)).
\(K\) is thus the special value that \(Q\) has when the reaction is at equilibrium
The value of Q in relation to K serves as an index how the composition of the reaction system compares to that of the equilibrium state, and thus it indicates the direction in which any net reaction must proceed. For example, if we combine the two reactants A and B at concentrations of 1 mol L –1 each, the value of Q will be 0÷1=0. The only possible change is the conversion of some of these reactants into products. If instead our mixture consists only of the two products C and D , Q will be indeterminately large (1÷0) and the only possible change will be in the reverse direction.
For example, if we combine the two reactants A and B at concentrations of 1 mol L –1 each, the value of Q will be 0÷1=0. The only possible change is the conversion of some of these reactants into products. If instead our mixture consists only of the two products C and D , Q will be indeterminately large (1÷0) and the only possible change will be in the reverse direction.
It is easy to see (by simple application of the Le Chatelier principle) that the ratio of Q/K immediately tells us whether, and in which direction, a net reaction will occur as the system moves toward its equilibrium state. A schematic view of this relationship is shown below:
It is very important that you be able to work out these relations for yourself, not by memorizing them, but from the definitions of \(Q\) and \(K\).
| Condition | Status of System |
|---|---|
| Q > K | Product concentration too high for equilibrium; net reaction proceeds to left . |
| Q = K | System is at equilibrium; no net change will occur. |
| Q < K | Product concentration too low for equilibrium; net reaction proceeds to right . |
It is very important that you be able to work out these relations for yourself, not by memorizing them, but from the definitions of \(Q\) and \(K\).
The equilibrium constant for the oxidation of sulfur dioxide is K p = 0.14 at 900 K.
\[\ce{2 SO_2(g) + O_2(g) \rightleftharpoons 2 SO_3(g)} \nonumber\]
If a reaction vessel is filled with SO 3 at a partial pressure of 0.10 atm and with O 2 and SO 2 each at a partial pressure of 0.20 atm, what can you conclude about whether, and in which direction, any net change in composition will take place?
Solution:
The value of the equilibrium quotient Q for the initial conditions is
\[ Q= \dfrac{p_{SO_3}^2}{p_{O_2}p_{SO_2}^2} = \dfrac{(0.10\; atm)^2}{(0.20 \;atm) (0.20 \; atm)^2} = 1.25\; atm^{-1} \nonumber\]
Since Q > K , the reaction is not at equilibrium, so a net change will occur in a direction that decreases Q . This can only occur if some of the SO 3 is converted back into products. In other words, the reaction will "shift to the left".
The formal definitions of Q and K are quite simple, but they are of limited usefulness unless you are able to relate them to real chemical situations. The following diagrams illustrate the relation between Q and K from various standpoints. Take some time to study each one carefully, making sure that you are able to relate the description to the illustration.
For the reaction
\[N_2O_{4(g)} \rightleftharpoons 2 NO_{2(g)} \nonumber\]
K c = 0.0059 at 298 K.
This equilibrium condition is represented by the red curve that passes through all points on the graph that satisfy the requirement that
\[Q = \dfrac{[NO_2]^2}{ [N_2O_4]} = 0.0059 \nonumber\]
There are of course an infinite number of possible Q 's of this system within the concentration boundaries shown on the plot. Only those points that fall on the red line correspond to equilibrium states of this system (those for which \(Q = K_c\)). The line itself is a plot of [NO 2 ] that we obtain by rearranging the equilibrium expression
\[[NO_2] = \sqrt{[N_2O_4]K_c} \nonumber\]
If the system is initially in a non-equilibrium state, its composition will tend to change in a direction that moves it to one that is on the line. Two such non-equilibrium states are shown. The state indicated by has \(Q > K\), so we would expect a net reaction that reduces Q by converting some of the NO 2 into N 2 O 4 ; in other words, the equilibrium "shifts to the left". Similarly, in state , Q < K , indicating that the forward reaction will occur.
The blue arrows in the above diagram indicate the successive values that Q assumes as the reaction moves closer to equilibrium. The slope of the line reflects the stoichiometry of the equation. In this case, one mole of reactant yields two moles of products, so the slopes have an absolute value of 2:1.
One of the simplest equilibria we can write is that between a solid and its vapor. In this case, the equilibrium constant is just the vapor pressure of the solid. Thus for the process
\[I_{2(s)} \rightleftharpoons I_{2(g)} \nonumber\]
all possible equilibrium states of the system lie on the horizontal red line and is independent of the quantity of solid present (as long as there is at least enough to supply the relative tiny quantity of vapor.)
So adding various amounts of the solid to an empty closed vessel (states and ) causes a gradual buildup of iodine vapor. Because the equilibrium pressure of the vapor is so small, the amount of solid consumed in the process is negligible, so the arrows go straight up and all lead to the same equilibrium vapor pressure.
The decomposition of ammonium chloride is a common example of a heterogeneous (two-phase) equilibrium. Solid ammonium chloride has a substantial vapor pressure even at room temperature:
\[NH_4Cl_{(s)} \rightleftharpoons NH_{3(g)} + HCl_{(g)}\]
Arrow traces the states the system passes through when solid NH 4 Cl is placed in a closed container. Arrow represents the addition of ammonia to the equilibrium mixture; the system responds by following the path back to a new equilibrium state which, as the Le Chatelier principle predicts, contains a smaller quantity of ammonia than was added. The unit slopes of the paths and reflect the 1:1 stoichiometry of the gaseous products of the reaction.


16.10: Introduction to Workplace Safety and Health
What you’ll learn to do: Describe the legal obligations for workplace safety and health
Workplace safety isn’t just a business best practice, it is, as OSHA posters declare, the law. Passed by Congress in 1970, the Occupational Safety and Health (OSH) Act created the Occupational Safety and Health Administration (OSHA), whose mission is “to assure safe and healthful working conditions for working men and women by setting and enforcing standards and by providing training, outreach, education and assistance.” [1] In this section, we’ll discuss employer requirements and employee rights under the OSH Act and associated state plans.
-
"
Occupational Safety and Health Act
." United States Department of Labor. Accessed August 20, 2019. ↵
Contributors and Attributions
- Construction helmets. Authored by : Pop and Zebra. Provided by : Unsplash. Located at : https://unsplash.com/photos/wp81DxKUd1E . License : CC0: No Rights Reserved . License Terms : Unsplash License


4.7: Relationships Between Genes, Genotypes and Phenotypes
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- Ying Liu
- City College of San Francisco
Terminology
A specific position along a chromosome is called a locus . Each gene occupies a specific locus (so the terms locus and gene are often used interchangeably). Each locus will have an allelic form (allele). The complete set of alleles (at all loci of interest) in an individual is its genotype . Typically, when writing out a genotype, only the alleles at the locus (loci) of interest are considered – all the others are present and assumed to be wild type. The visible or detectable effect of these alleles on the structure or function of that individual is called its phenotype – what it looks like. The phenotype studied in any particular genetic experiment may range from simple, visible traits such as hair color, to more complex phenotypes including disease susceptibility or behavior. If two alleles are present in an individual, then various interactions between them may influence their expression in the phenotype.
Complete Dominance
Let us return to an example of a simple phenotype: flower color in Mendel’s peas. We have already said that one allele as a homozygote produces purple flowers, while the other allele as a homozygote produces white flowers (see Figures 1.8 and 3.3). But what about an individual that has one purple allele and one white allele; what is the phenotype of an individual whose genotype is heterozygous? This can only be determined by experimental observation. We know from observation that individuals heterozygous for the purple and white alleles of the flower color gene have purple flowers. Thus, the allele associated with purple color is therefore said to be dominant to the allele that produces the white color. The white allele, whose phenotype is masked by the purple allele in a heterozygote, is recessive to the purple allele.
To represent this relationship, often, a dominant allele will be represented by a capital letter (e.g. A ) while a recessive allele will be represented in lower case (e.g. a ). However, many different systems of genetic symbols are in use. The most common are shown in Table \(\PageIndex{1}\). Also note that genes and alleles are usually written in italics and chromosomes and proteins are not. For example, the white gene in Drosophila melanogaster on the X chromosome encodes a protein called WHITE.
|
Examples |
Interpretation |
|---|---|
|
A and a |
Uppercase letters represent dominant alleles and lowercase letters indicate recessive alleles. Mendel invented this system but it is not commonly used because not all alleles show complete dominance and many genes have more than two alleles. |
|
a + and a 1 |
Superscripts or subscripts are used to indicate alleles. For wild type alleles the symbol is a superscript +. |
|
AA or A/A |
Sometimes a forward slash is used to indicate that the two symbols are alleles of the same gene, but on homologous chromosomes. |
Incomplete Dominance
Besides dominance and recessivity, other relationships can exist between alleles. In incomplete dominance (also called semi-dominance, Figure \(\PageIndex{4}\)), both alleles affect the trait additively, and the phenotype of the heterozygote is intermediate between either of the homozygotes. For example, alleles for color in carnation flowers (and many other species) exhibit incomplete dominance. Plants with an allele for red petals ( A 1 ) and an allele for white petals ( A 2 ) have pink petals. We say that the A 1 and the A 2 alleles show incomplete dominance because neither allele is completely dominant over the other.
Co-Dominance
Co-dominance is another type of allelic relationship, in which a heterozygous individual expresses the phenotype of both alleles simultaneously. An example of co-dominance is found within the ABO blood group of humans. The ABO gene has three common alleles which were named (for historical reasons) I A , I B , and i . People homozygous for I A or I B display only A or B type antigens, respectively, on the surface of their blood cells, and therefore have either type A or type B blood (Figure \(\PageIndex{5}\)). Heterozygous I A I B individuals have both A and B antigens on their cells, and so have type AB blood. Notice that the heterozygote expresses both alleles simultaneously, and is not some kind of novel intermediate between A and B. Co-dominance is therefore distinct from incomplete dominance, although they are sometimes confused.
Figure \(\PageIndex{5}\): Relationship between genotype and phenotype for three alleles of the human ABO gene. The I A and I B alleles show co-dominance. The I A allele is completely dominant to the i allele. The I B allele is completely dominant to the i allele. (Original-Deholos -CC:AN)It is also important to note that the third allele, i , does not make either antigen and is recessive to the other alleles. I A /i or I B /i individuals display A or B antigens respectively. People homozygous for the i allele have type O blood. This is a useful reminder that different types of dominance relationships can exist, even for alleles of the same gene. Many types of molecular markers, which we will discuss in a later chapter, display a co-dominant relationship among alleles.


35: Appendix A - Lab Safety
The following guidelines were designed by the Biology Department Faculty for use in all biology courses offered at Evergreen Valley College. It is mandatory that students follow these regulations. Students are expected to read these regulations, sign the Safety Form on page 5, and return the bottom half to the instructor. This procedure must be completed before a student can begin any laboratory activity. The student should keep the regulations for future reference.
General*
- Read all laboratory instructions before coming to class.
- Dress appropriately for lab: closed-toe shoes, long hair tied back, no dangling jewelry or loose clothing. Avoid wearing contact lenses.
 Keep the safety equipment and exits free of obstruction.
- Behave in a professional manner.
- Only registered students are allowed in the laboratory.
- Students are not permitted access to prep rooms, laboratory cupboards, or instructor desks without prior approval from the instructor.
- Students who have any medical conditions (e.g. diabetes, immunological defect, pregnancy, epilepsy) that may necessitate special precautions in the laboratory must inform the professor.
- To encourage classroom unity and promote safety, please speak English in class.
- No food or drink (including water) allowed in the laboratory.
- Keep hands away from face, eyes and clothing while using hazardous solutions or materials. Do not handle contact lenses or apply cosmetics during class.
- Maintain a clean, open work area free of everything except material related to the exercise.
- Use all equipment and materials with care and as authorized by instructor.
- Report all spills or accidents to the instructor. Clean up spills immediately.
- Wipe down work surfaces at the beginning and end of each lab.
- Wash your hands at the beginning and end of class.
- At the end of lab make sure all items are returned to their proper location. Push in chairs. Make sure sinks and your work space are clean and free of debris.
*Covid precautions
- Perform a Self-Check for the following symptoms listed below prior to coming to lab. If any symptoms are present, do not come to lab.
- Fever or chills, cough, shortness of breath or difficulty breathing, fatigue, muscle or body aches, new loss of taste or smell, sore throat, congestion or runny nose, nausea or vomiting, diarrhea
- Wear a mask at all times, regardless of vaccination status.
- Maintain social distancing, when possible.
Emergency Preparedness
- During an earthquake stay in the room and immediately get under the desk. After the earthquake follow your instructor’s directions.
- During a fire or other emergencies requiring evacuation, meet at the designated evacuation location. Students are required to meet and remain at the evacuation site until dismissed by the instructor.
- During a shelter-in-place emergency, students should remain in the classroom and follow instructions.
Handling Equipment and Materials
- Properly label all glassware and slides.
- Keep all equipment and materials away from the edge of the lab benches.
- Use protective aids (e.g. test-tube clamps, hot pads) when handling hot glassware.
- Light gas burners only as instructed. Be sure no volatile or flammable materials are being used nearby. Remember to have long hair tied back, no dangling jewelry or loose clothing.
- When heating a test tube do not look into the mouth of the tube or point it into the direction of any person.
- Heat sources should only be on when in use. Do not leave heat sources unattended. When in doubt, students should assume a heat source is on or hot.
- Operate electrical equipment in a dry area with dry hands.
- When removing an electrical plug from its socket, pull the plug, not the cord.
- Inform your instructor immediately if equipment or materials are broken or not working properly.
- Never return reagents back into their original containers. Keep containers capped when not in use.
Handling Hazardous Material
- Wear appropriate eye protection as directed by the instructor.
- Wear disposable gloves when handling hazardous materials including chemicals, organisms, or bodily fluids.
- Wear protective clothing when required by the instructor.
- Follow correct disposal procedures as detailed in “Cleanup and Disposal.”
- To prevent contamination by needle stick injuries, always use new needles and lancets.
- Use the fume hood whenever noxious, corrosive or toxic fumes are produced or released.
Bacteria and Fungi
- Never open Petri dishes containing bacterial or fungal growth unless directed to do so by the instructor.
- Dispose of all discarded bacterial and fungal cultures as directed by the instructor.
Cleanup and Disposal
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Follow correct disposal procedures – dispose of used materials promptly and in the appropriate disposal containers.
- Dispose of broken glass in labeled glass disposal containers.
- Dispose of needles and lancets in puncture-proof sharp containers.
- Dispose of used chemicals in labeled and capped waste containers as directed by your instructor.
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For disposal of items contaminated with organisms or bodily fluids,
- Decontaminate slides, spectrophotometer tubes, and other glassware in 10% bleach.
- Dispose of soft items (i.e. gauze or swabs) in labeled biohazard containers.
- Dispose of all animal tissues in labeled biohazard containers.
- Place micropipette tips, sample vials, and contaminated glassware in a labeled waste container or designated area.
- See instructor for disposal of cadaver tissues.
Safety Pledge
I have read the Biology Laboratory Safety Guidelines and understand the rules and procedures of proper safety in the biology laboratory course listed below. I recognize my responsibility and pledge to observe all safety rules in the biology laboratory at all times.
I acknowledge that the instructor has covered all safety guidelines and equipment that we may need to use this semester.
Name of Student (please print)_____________________________________________________
Signature of Student _______________________________________________
Date ___________________
Course ___________________ Section _______________ Professor ____________________________
Student copy
Safety Pledge
I have read the Biology Laboratory Safety Guidelines and understand the rules and procedures of proper safety in the biology laboratory course listed below. I recognize my responsibility and pledge to observe all safety rules in the biology laboratory at all times.
I acknowledge that the instructor has covered all safety guidelines and equipment that we may need to use this semester.
Name of Student (please print)_____________________________________________________
Signature of Student _______________________________________________
Date ___________________
Course ___________________ Section _______________ Professor ____________________________
Instructor copy


5.26: Skills Checklist- Assisting With a Bedpan
- Gather Supplies: Gloves, bedpan, barrier, and toilet tissue
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Routine Pre-Procedure Steps:
- Knock on the client’s door.
- Perform hand hygiene.
- Maintain respectful, courteous, and professional communication at all times.
- Introduce yourself and identify the resident.
- Provide for privacy.
- Explain the procedure to the client.
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Procedure Steps:
- Put on gloves.
- Turn the resident or raise their hips and place a barrier (e.g., a towel, waterproof soaker pad, disposable pad) under their buttocks.
- Position the resident on the bedpan/fracture pan correctly. The deeper portion of the bedpan should be directed toward their toes, and the resident should be centered on the bedpan. For fracture bedpans, the handle should be directed toward their toes.
- Raise the head of the bed to a comfortable level.
- Cover the resident with linens or a bath blanket.
- Leave toilet tissue within reach of the resident.
- Leave the call light within reach of the resident.
- Wait nearby allowing for resident privacy.
- When the resident signals, return and assist the resident to perform hand hygiene.
- Discard the soiled linen in the designated laundry hamper.
- Gently remove the bedpan/fracture pan.
- Assist with perineal care.
- Empty the bedpan into the toilet or into a graduated cylinder if output is being recorded. Note the amount and characteristics (i.e., color, clarity, sediment, or unusual odor) of the urine. Empty the urine from the graduated cylinder used into the toilet.
- Rinse the equipment used and empty the rinse water into the toilet.
- Remove the gloves, turning them inside out.
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Post-Procedure Steps:
- Perform hand hygiene.
- Check for resident comfort and ask if anything else is needed.
- Ensure the bed is low and locked. Check the brakes.
- Place the call light or signaling device within reach of the resident.
- Open the door and privacy curtain.
- Perform hand hygiene.
- Document urinary output in mL and report any issues or changes noted with the resident.
View a YouTube video [1] of an instructor demonstrating a ssisting with a bedpan:
- Chippewa Valley Technical College. (2022, December 3). Assisting with a Bedpan. [Video]. YouTube. Video licensed under CC BY 4.0 . https://youtu.be/WCg57s4HGhc ↵


1.3: Basics of Singing 2- Warmups and Practice Technique
Warming up the voice before singing a song is similar to warming up before an athletic workout. Sure, you can work out without warming up, but the effectiveness of the workout is diminished, and there is a better chance of straining a muscle. Since singing is a physical exercise, it should be approached no differently. There are different schools of thought when approaching the length of time a singer should warm up. No less than 5 minutes is advised, but too much more may not be productive. The warm up should also fit what the singer plans on practicing. If the song has a difficult run, then singing a warm up similar to the run will ready the voice for that passage. Remember, the goal is to eliminate any extra tension in the voice, and this gives the singer muscle memory of what is to be practiced.
There are several components to a healthy warm up:
1. Physical stretching;
2. Breathing exercise;
3. Single note sustained on a vowel;
4. Simple narrow range exercise;
5. Articulator exercise;
6. Range extension; and
7. Longer, wider, faster.
Physical stretching allows for the release of any tension in the body, and warms up the muscles needed for singing. One area to focus on relaxing is the neck, where a lot of tension is naturally held and problematic for singing. When stretching the neck, do not tilt your head backwards, or roll backwards, as that can counteract the release of tension. During stretching, be aware of your posture. Remember, the least number of muscle groups engaged equals less tension on the body and voice.
For the breathing exercise, a suggestion is to simply breathe in fully and out, focusing on a deep, low breath.  A slow count of 4 seconds in and out is a good rule. This also provides muscle memory for the type of breathing necessary to sing. Begin then to breathe in on 4, and hiss the air out in multiples of 4 (8 seconds, then 12, etc.). Go through at least 4 sequences of these breath exercises. While hissing, place one hand on the “pant muscle” and the other around the bottom of your rib cage on the side. Your hand on the “pant muscle” should feel it fully engaged throughout the hiss, and the hand on your side can act as a guide to expand your breath. Besides a hiss, other types of sounds will work, such as “sh” or “f.”
Next, sing a single note in the middle of your range (a G is good if you are familiar with music notation) on a vowel. The “oo” vowel is a good one to start with as it is easier to keep round, tall, and free of tension. Try and hold the note also for counts in multiples of 4, changing the note higher or lower for a few steps as you increase the time.
A small range of notes is a common type of warm up. There are many types out there to try, so I will give a couple here. Usually, it is a five note scale going down (5,4,3,2,1 or G,F,E,D,C or sol fa mi re do, depending on your thought process). The exercise should be done with one consonant and one vowel, for example Na Na Na Na Na). You then move up a step and continue the same exercise, starting from the middle of your range up to a medium high part of your range. Then do a different vowel and constant starting from the middle of your range going down to a medium low part of your range. What vowel and consonant to pick varies on what you are working on in your song practice that day.
The next exercise works your articulators: your tongue, teeth, and lips. One of the great advantages singers have over instrumental musicians is the ability to communicate through words. Unfortunately, this can backfire if the audience cannot understand the words. Being aware of how to use your articulators differently in singing versus talking is imperative if the singer wants to be an effective communicator. One example of an exercise is a five-note scale going up and down using “ming” (holding the ng at the end of each one). Other examples are “flah,” “zing,” “blah,” “ping,” and “bah lah.” Again, which ones to use should be reflected by the music that will follow the warm up.
Range extension exercises can be tricky for the inexperienced singer, so when beginning these, be careful not to go too high too fast. One of the greatest difficulties singers encounter with tone is tension in the high parts of the voice.  We will talk more about how to avoid this in later lessons, but for now, remember to always do three things when singing high: 1. Open the throat in a yawn 2. Keep the diaphragm engaged (check that “pant muscle”) and 3. Approach with thinking “top-down” and not “up.” An example of a range extension exercise is an arpeggio up and down (C-E-G-C-G-E-C or 1-3-5-8-5-3-1 or do-mi-sol-do-sol-mi-do) on a vowel-consonant combination such as “nah.” You can also use different vowel-consonant combinations, or words (“I sigh to sing” or “O row the boat.”) Sing the arpeggio going up by step to the top and then bottom of your comfortable range. A rule to follow is to sing this exercise a couple notes higher or lower than your highest/lowest note in the song about to be practiced.
Longer, wider, faster refers to the idea of singing exercises that combine all the previous ones into an exercise that imitates phrases in actual music. When we sing music, typically there is a musical idea, or phrase, which should be sung in one breath. This exercise prepares the singer for that final transfer to singing a song. One example uses a vowel “ah” jumping from a low to high octave (C-C, 1-8, do-high do), singing a turn and then a scale passage down (C-C, BCDBAGFEDC, 1-8,<PHONE_NUMBER>1, do-do, ti do re do ti la sol fa me re do). You would then go up a step and add a turn (C-C, BCDC BCDCBAGFEDC, etc). Each turn would lengthen the phrase. There are several variations on this exercise, but start there.
In essence, the warm up should not be skipped to simply start practicing the song because it is “boring.” You should change the type of syllables each warm up to reflect the song, again reinforcing good muscle memory habits for the practice to follow. Think of it as a way to reset your voice back to true voice before beginning to sing the song. Without this step, the bad habits formed before this course will continue to occur.
Supplemental Videos
Assignment on Videos
After watching the demonstration video, enter the secret number at the top of your assignment.
Write in complete sentences answers to the following questions.
1. In your own words, what was the content of the video?
2. What are two things you found most interesting about the content of the video?
3. Think of a singer you have seen and heard. Who are they, and what do they demonstrate in terms of this concept?
4. Name 3 positive things you do while singing that relate to this concept.
5. What 2 things can you improve on relating to this concept?