6.10.1E: Examples and Elementary Properties Exercises

Exercises for 1

Exercise \(\PageIndex{1}\)

Show that each of the following functions is a linear transformation.\(T : \mathbb{R}^2 \to \mathbb{R}^2\); \(T(x, y) = (x, -y)\) (reflection in the \(x\) axis)

1. \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 ; T(x, y)=(x,-y)\) (reflection in the \(x\) axis)
2. \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 ; T(x, y, z)=(x, y,-z)\) (reflection in the \(x-y\) plane)
3. \(T: \mathbb{C} \rightarrow \mathbb{C} ; T(z)=\bar{z}\) (conjugation)
4. \(T: \mathbf{M}_{m n} \rightarrow \mathbf{M}_{k l} ; T(A)=P A Q, P\) a \(k \times m\) matrix, \(Q\) an \(n \times l\) matrix, both fixed
5. \(T: \mathbf{M}_{n n} \rightarrow \mathbf{M}_{n n} ; T(A)=A^T+A\)
6. \(T: \mathbf{P}_n \rightarrow \mathbb{R} ; T[p(x)]=p(0)\)
7. \(T: \mathbf{P}_n \rightarrow \mathbb{R} ; T\left(r_0+r_1 x+\ldots+r_n x^n\right)=r_n\)
8. \(T: \mathbb{R}^n \rightarrow \mathbb{R} ; T(\mathbf{x})=\mathbf{x} \cdot \mathbf{z}, \mathbf{z}\) a fixed vector in \(\mathbb{R}^n\)
9. \(T: \mathbf{P}_n \rightarrow \mathbf{P}_n ; T[p(x)]=p(x+1)\)
10. \(T: \mathbb{R}^n \rightarrow V ; T\left(r_1, \ldots, r_n\right)=r_1 \mathbf{e}_1+\ldots+r_n \mathbf{e}_n\) where \(\left\{\mathbf{e}_1, \ldots, \mathbf{e}_n\right\}\) is a fixed basis of \(V\).
11. \(T: V \rightarrow \mathbb{R} ; T\left(r_1 \mathbf{e}_1+\ldots+r_n \mathbf{e}_n\right)=r_1\), where \(\left\{\mathbf{e}_1, \ldots, \mathbf{e}_n\right\}\) is a fixed basis of \(V\)

Answer

2. \(T(\vec{v})=\vec{v} A\) where \(A=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right]\)
3. \(T(A+B)=P(A+B) Q=P A Q+P B Q=T(A)+T(B) ; T(r A)=P(r A) Q=r P A Q=r T(A)\)
4. \[\begin{array}{l}T[(p+q)(x)]=(p+q)(0)=p(0)+q(0)=T[p(x)]+T[q(x)] \\T[(r p)(x)]=(r p)(0)=r(p(0))=r T[p(x)]\end{array}\]
5. \(T(X+Y)=(X+Y) \cdot Z=X \cdot Z+Y \cdot Z=T(X)+T(Y)\), and \(T(r X)=(r X) \cdot Z=r(X \cdot Z)=r T(X)\)
6. If \(\vec{v}=\left(v_1, \ldots, v_n\right)\) and \(\vec{w}=\left(w_1, \ldots, w_n\right)\), then \(T(\vec{v}+\vec{w})=\left(v_1+w_1\right)\vec{e}_1+\cdots+\left(v_n+w_n\right) \vec{e}_n=\left(v_1 \vec{e}_1+\cdots+v_n\vec{e}_n\right)+\left(w_1 \vec{e}_1+\cdots+w_n \vec{e}_n\right)=T(\vec{v})+T(\vec{w})\) \(T(a \vec{v})=\left(a v_1\right)\vec{e}+\cdots+\left(a v_n\right) \vec{e}_n=a\left(v \vec{e}+\cdots+v_n \vec{e}_n\right)=a T(\vec{v})\)

Exercise \(\PageIndex{2}\)

In each case, show that \(T\) is not a linear transformation.

1. \(T : \mathbf{M}_{nn} \to \mathbb{R}\); \(T(A) = \det A\)
2. \(T : \mathbf{M}_{nm} \to \mathbb{R}\); \(T(A) = rank \mathbf{A}\)
3. \(T : \mathbb{R} \to \mathbb{R}\); \(T(x) = x^{2}\)
4. \(T : V \to V\); \(T(\mathbf{v}) = \mathbf{v} + \mathbf{u}\) where \(\mathbf{u} \neq \mathbf{0}\) is a fixed vector in \(V\) (\(T\) is called the translation by \(\mathbf{u}\))

Answer

2. \( rank \;\((A+B) \neq \backslash\) rank \; \(A+\backslash\) rank \; \(B\) in general. For example, \(A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) and \(B=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]\)
3. \(T(\overrightarrow{0})=\overrightarrow{0}+\vec{u}=\vec{u} \neq \overrightarrow{0}\), so \(T\) is not linear by Theorem [thm:020817].

Exercise \(\PageIndex{3}\)

In each case, assume that \(T\) is a linear transformation.

1. If \(T : V \to \mathbb{R}\) and \(T(\mathbf{v}_{1}) = 1\), \(T(\mathbf{v}_{2}) = -1\), find \(T(3\mathbf{v}_{1} - 5\mathbf{v}_{2})\).
2. If \(T : V \to \mathbb{R}\) and \(T(\mathbf{v}_{1}) = 2\), \(T(\mathbf{v}_{2}) = -3\), find \(T(3\mathbf{v}_{1} + 2\mathbf{v}_{2})\).

3. If \(T : \mathbb{R}^2 \to \mathbb{R}^2\) and \(T\left[ \begin{array}{r} 1 \\ 3 \end{array} \right] = \left[ \begin{array}{r} 1 \\ 1 \end{array} \right]\),
   \(T\left[ \begin{array}{r} 1 \\ 1 \end{array} \right] = \left[ \begin{array}{r} 0 \\ 1 \end{array} \right]\), find \(T\left[ \begin{array}{r} -1 \\ 3 \end{array} \right]\).

4. If \(T : \mathbb{R}^2 \to \mathbb{R}^2\) and \(T\left[ \begin{array}{r} 1 \\ -1 \end{array} \right] = \left[ \begin{array}{r} 0 \\ 1 \end{array} \right]\),
   \(T\left[ \begin{array}{r} 1 \\ 1 \end{array} \right] = \left[ \begin{array}{r} 1 \\ 0 \end{array} \right]\), find \(T\left[ \begin{array}{r} 1 \\ -7 \end{array} \right]\).

5. If \(T : \mathbf{P}_{2} \to \mathbf{P}_{2}\) and \(T(x + 1) = x\), \(T(x - 1) = 1\), \(T(x^{2}) = 0\), find \(T(2 + 3x - x^{2})\).
6. If \(T : \mathbf{P}_{2} \to \mathbb{R}\) and \(T(x + 2) = 1\), \(T(1) = 5\), \(T(x^{2} + x) = 0\), find \(T(2 - x + 3x^{2})\).

Answer

2. \(T\left(3 \vec{v}_1+2 \vec{v}_2\right)=0\)
3. \(T\left[\begin{array}{r}1 \\ -7\end{array}\right]=\left[\begin{array}{r}-3 \\ 4\end{array}\right]\)
4. \(T\left(2-x+3 x^2\right)=46\)

Exercise \(\PageIndex{4}\)

In each case, find a linear transformation with the given properties and compute \(T(\mathbf{v})\).

1. \(T : \mathbb{R}^2 \to \mathbb{R}^3\); \(T(1, 2) = (1, 0, 1)\),
   \(T(-1, 0) = (0, 1, 1)\); \(\mathbf{v} = (2, 1)\)

2. \(T : \mathbb{R}^2 \to \mathbb{R}^3\); \(T(2, -1) = (1, -1, 1)\),
   \(T(1, 1) = (0, 1, 0)\); \(\mathbf{v} = (-1, 2)\)

3. \(T : \mathbf{P}_{2} \to \mathbf{P}_{3}\); \(T(x^{2}) = x^{3}\), \(T(x + 1) = 0\),
   \(T(x - 1) = x\); \(\mathbf{v} = x^{2} + x + 1\)

4. \(T : \mathbf{M}_{22} \to \mathbb{R}\); \(T\left[ \begin{array}{rr} 1 & 0 \\ 0 & 0 \end{array} \right] = 3\), \(T\left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right] = -1\), \(T\left[ \begin{array}{rr} 1 & 0 \\ 1 & 0 \end{array} \right] = 0 = T\left[ \begin{array}{rr} 0 & 0 \\ 0 & 1 \end{array} \right]\); \(\mathbf{v} = \left[ \begin{array}{rr} a & b \\ c & d \end{array} \right]\)

Answer

2. \(T(x, y)=\frac{1}{3}(x-y, 3 y, x-y) ; T(-1,2)=(-1,2,-1)\)
3. \(T\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=3 a-3 c+2 b\)

Exercise \(\PageIndex{5}\)

If \(T : V \to V\) is a linear transformation, find \(T(\mathbf{v})\) and \(T(\mathbf{w})\) if:

1. \(T(\mathbf{v} + \mathbf{w}) = \mathbf{v} - 2\mathbf{w}\) and \(T(2\mathbf{v} - \mathbf{w}) = 2\mathbf{v}\)
2. \(T(\mathbf{v} + 2\mathbf{w}) = 3\mathbf{v} - \mathbf{w}\) and \(T(\mathbf{v} - \mathbf{w}) = 2\mathbf{v} - 4\mathbf{w}\)

Answer

b. \(T(\mathbf{v}) = \frac{1}{3}(7\mathbf{v} - 9\mathbf{w})\), \(T(\mathbf{w}) = \frac{1}{3}(\mathbf{v} + 3\mathbf{w})\)

Example \(\PageIndex{6}\)

If \(T : V \to W\) is a linear transformation, show that \(T(\mathbf{v} - \mathbf{v}_{1}) = T(\mathbf{v}) - T(\mathbf{v}_{1})\) for all \(\mathbf{v}\) and \(\mathbf{v}_{1}\) in \(V\).

Example \(\PageIndex{7}\)

Let \(\{\mathbf{e}_{1}, \mathbf{e}_{2}\}\) be the standard basis of \(\mathbb{R}^2\). Is it possible to have a linear transformation \(T\) such that \(T(\mathbf{e}_{1})\) lies in \(\mathbb{R}\) while \(T(\mathbf{e}_{2})\) lies in \(\mathbb{R}^2\)? Explain your answer.

Exercise \(\PageIndex{8}\)

Let \(\{\mathbf{v}_{1}, \dots, \mathbf{v}_{n}\}\) be a basis of \(V\) and let \(T : V \to V\) be a linear transformation.

1. If \(T(\mathbf{v}_{i}) = \mathbf{v}_{i}\) for each \(i\), show that \(T = 1_{V}\).
2. If \(T(\mathbf{v}_{i}) = -\mathbf{v}_{i}\) for each \(i\), show that \(T = -1\) is the scalar operator (see Example \(\PageIndex{1}\)).

Answer

b. T(\vec{v})=(-1) \vec{v} \text { for all } \vec{v} \text { in } V \text {, so } T \text { is the scalar operator }-1 \text {. }

Example \(\PageIndex{9}\)

If \(A\) is an \(m \times n\) matrix, let \(C_{k}(A)\) denote column \(k\) of \(A\). Show that \(C_{k} : \mathbf{M}_{mn} \to \mathbb{R}^m\) is a linear transformation for each \(k = 1, \dots, n\).

Example \(\PageIndex{10}\)

Let \(\{\mathbf{e}_{1}, \dots, \mathbf{e}_{n}\}\) be a basis of \(\mathbb{R}^n\). Given \(k\), \(1 \leq k \leq n\), define \(P_{k} : \mathbb{R}^n \to \mathbb{R}^n\) by
\(P_{k}(r_{1}\mathbf{e}_{1} + \cdots + r_{n}\mathbf{e}_{n}) = r_{k}\mathbf{e}_{k}\). Show that \(P_{k}\) a linear transformation for each \(k\).

Example \(\PageIndex{11}\)

Let \(S : V \to W\) and \(T : V \to W\) be linear transformations. Given \(a\) in \(\mathbb{R}\), define functions
\((S + T) : V \to W\) and \((aT) : V \to W\) by \((S + T)(\mathbf{v}) = S(\mathbf{v}) + T(\mathbf{v})\) and \((aT)(\mathbf{v}) = aT(\mathbf{v})\) for all \(\mathbf{v}\) in \(V\). Show that \(S + T\) and \(aT\) are linear transformations.

Exercise \(\PageIndex{12}\)

Describe all linear transformations \(T : \mathbb{R} \to V\).

Answer

\text { If } T(1)=\vec{v} \text {, then } T(r)=T(r \cdot 1)=r T(1)=r \vec{v} \text { for all } r \text { in } \mathbb{R} \text {. }

Example \(\PageIndex{13}\)

Let \(V\) and \(W\) be vector spaces, let \(V\) be finite dimensional, and let \(\mathbf{v} \neq \mathbf{0}\) in \(V\). Given any \(\mathbf{w}\) in \(W\), show that there exists a linear transformation \(T : V \to W\) with \(T(\mathbf{v}) = \mathbf{w}\). [Hint: Theorem \(\PageIndex{3}\) and Theorem 6.4.1.]

Example \(\PageIndex{14}\)

Given \(\mathbf{y}\) in \(\mathbb{R}^n\), define \(S_{\mathbf{y}} : \mathbb{R}^n \to \mathbb{R}\) by \(S_{\mathbf{y}}(\mathbf{x}) = \mathbf{x} \cdot \mathbf{y}\) for all \(\mathbf{x}\) in \(\mathbb{R}^n\) (where \(\cdot\) is the dot product introduced in Section 5_3).

1. Show that \(S_{\mathbf{y}} : \mathbb{R}^n \to \mathbb{R}\) is a linear transformation for any \(\mathbf{y}\) in \(\mathbb{R}^n\).
2. Show that every linear transformation \(T : \mathbb{R}^n \to \mathbb{R}\) arises in this way; that is, \(T = S_{\mathbf{y}}\) for some \(\mathbf{y}\) in \(\mathbb{R}^n\). [Hint: If \(\{\mathbf{e}_{1}, \dots, \mathbf{e}_{n}\}\) is the standard basis of \(\mathbb{R}^n\), write \(S_{\mathbf{y}}(\mathbf{e}_{i}) = y_{i}\) for each \(i\). Use Theorem \(\PageIndex{1}\).]

Exercise \(\PageIndex{15}\)

Let \(T : V \to W\) be a linear transformation.

1. If \(U\) is a subspace of \(V\), show that
   \(T(U) = \{T(\mathbf{u}) \mid \mathbf{u} \mbox{ in } U\}\) is a subspace of \(W\) (called the image of \(U\) under \(T\)).

2. If \(P\) is a subspace of \(W\), show that
   \(\{\mathbf{v} \mbox{ in } V \mid T(\mathbf{v}) \mbox{ in } P\}\) is a subspace of \(V\) (called the preimage of \(P\) under \(T\)).

Answer

b. \(\mathbf{0}\) is in \(U = \{\mathbf{v} \in V \mid T(\mathbf{v}) \in P\}\) because \(T(\mathbf{0}) = \mathbf{0}\) is in \(P\). If \(\mathbf{v}\) and \(\mathbf{w}\) are in \(U\), then \(T(\mathbf{v})\) and \(T(\mathbf{w})\) are in \(P\). Hence \(T(\mathbf{v} + \mathbf{w}) = T(\mathbf{v}) + T(\mathbf{w})\) is in \(P\) and \(T(r\mathbf{v}) = rT(\mathbf{v})\) is in \(P\), so \(\mathbf{v} + \mathbf{w}\) and \(r\mathbf{v}\) are in \(U\).

Example \(\PageIndex{16}\)

Show that differentiation is the only linear transformation \(\mathbf{P}_{n} \to \mathbf{P}_{n}\) that satisfies \(T(x^{k}) = kx^{k-1}\) for each \(k = 0, 1, 2, \dots, n\).

Example \(\PageIndex{17}\)

Let \(T : V \to W\) be a linear transformation and let \(\mathbf{v}_{1}, \dots, \mathbf{v}_{n}\) denote vectors in \(V\).

1. If \(\{T(\mathbf{v}_{1}), \dots, T(\mathbf{v}_{n})\}\) is linearly independent, show that \(\{\mathbf{v}_{1}, \dots, \mathbf{v}_{n}\}\) is also independent.
2. Find \(T : \mathbb{R}^2 \to \mathbb{R}^2\) for which the converse of part (a) is false.

Exercise \(\PageIndex{18}\)

Suppose \(T : V \to V\) is a linear operator with the property that \(T\left[T(\mathbf{v})\right] = \mathbf{v}\) for all \(\mathbf{v}\) in \(V\). (For example, transposition in \(\mathbf{M}_{nn}\) or conjugation in \(\mathbb{C}\).) If \(\mathbf{v} \neq \mathbf{0}\) in \(V\), show that \(\{\mathbf{v}, T(\mathbf{v})\}\) is linearly independent if and only if \(T(\mathbf{v}) \neq \mathbf{v}\) and \(T(\mathbf{v}) \neq -\mathbf{v}\).

Answer

Suppose \(r \vec{v}+s T(\vec{v})=\overrightarrow{0}\). If \(s=0\), then \(r=0\) (because \(\vec{v} \neq \overrightarrow{0}\) ). If \(s \neq 0\), then \(T(\vec{v})=a \vec{v}\) where \(a=-s^{-1} r\). Thus \(\vec{v}=T^2(\vec{v})=T(a \vec{v})=a^2 \vec{v}\), so \(a^2=1\), again because \(\vec{v} \neq \overrightarrow{0}\). Hence \(a= \pm 1\). Conversely, if \(T(\vec{v})= \pm \vec{v}\), then \(\{\vec{v}, T(\vec{v})\}\) is certainly not independent.

Example \(\PageIndex{19}\)

If \(a\) and \(b\) are real numbers, define \(T_{a,b} : \mathbb{C} \to \mathbb{C}\) by \(T_{a,b}(r + si) = ra + sbi\) for all \(r + si\) in \(\mathbb{C}\).

1. Show that \(T_{a,b}\) is linear and \(T_{a,b}(\overline{z}) = \overline{T_{a,b}(z)}\) for all \(z\) in \(\mathbb{C}\). (Here \(\overline{z}\) denotes the conjugate of \(z\).)
2. If \(T : \mathbb{C} \to \mathbb{C}\) is linear and \(T(\overline{z}) = \overline{T(z)}\) for all \(z\) in \(\mathbb{C}\), show that \(T = T_{a,b}\) for some real \(a\) and \(b\).

Example \(\PageIndex{20}\)

Show that the following conditions are equivalent for a linear transformation \(T : \mathbf{M}_{22} \to \mathbf{M}_{22}\).

1. \(tr \mathbf{\left[T(A)\right]} = tr \mathbf{A}\) for all \(A\) in \(\mathbf{M}_{22}\).

2. \(T\left[ \begin{array}{cc} r_{11} & r_{12} \\ r_{21} & r_{22} \end{array} \right] = r_{11}B_{11} + r_{12}B_{12} + r_{21}B_{21} + r_{22}B_{22}\) for matrices \(B_{ij}\) such that
   \(tr \mathbf{B_{11}} = 1 = tr \mathbf{B_{22}}\) and \(tr \mathbf{B_{12}} = 0 = tr \mathbf{B_{21}}\).

Exercise \(\PageIndex{21}\)

Given \(a\) in \(\mathbb{R}\), consider the evaluation map \(E_{a} : \mathbf{P}_{n} \to \mathbb{R}\) defined in Example [exa:020790].

1. Show that \(E_{a}\) is a linear transformation satisfying the additional condition that \(E_{a}(x^{k}) = \left[E_{a}(x)\right]^{k}\) holds for all \(k = 0, 1, 2, \dots\). [Note: \(x^{0} = 1\).]
2. If \(T : \mathbf{P}_{n} \to \mathbb{R}\) is a linear transformation satisfying \(T(x^{k}) = \left[T(x)\right]^{k}\) for all \(k = 0, 1, 2, \dots\), show that \(T = E_{a}\) for some \(a\) in \(R\).

Answer

b. Given such a \(T\), write \(T(x) = a\). If \(p = p(x) = \sum_{i=0}^{n}a_{i}x^{i}\), then \(T(p) = \sum a_{i}T(x^i) = \sum a_i\left[T(x)\right]^i = \sum a_{i}a^i = p(a) = E_a(p)\). Hence \(T = E_a\)

Example \(\PageIndex{22}\)

If \(T : \mathbf{M}_{nn} \to \mathbb{R}\) is any linear transformation satisfying \(T(AB) = T(BA)\) for all \(A\) and \(B\) in \(\mathbf{M}_{nn}\), show that there exists a number \(k\) such that \(T(A) = k tr \mathbf{A}\) for all \(A\). (See Lemma 5.5.1.) [Hint: Let \(E_{ij}\) denote the \(n \times n\) matrix with \(1\) in the \((i, j)\) position and zeros elsewhere.

Show that \(E_{ik}E_{lj} = \left\lbrace \begin{array}{cl} 0 & \mbox{if } k \neq l \\ E_{ij} & \mbox{if } k = l \end{array} \right.\). Use this to show that \(T(E_{ij}) = 0\) if \(i \neq j\) and
\(T(E_{11}) = T(E_{22}) = \cdots = T(E_{nn})\). Put \(k = T(E_{11})\) and use the fact that \(\{E_{ij} \mid 1 \leq i, j \leq n\}\) is a basis of \(\mathbf{M}_{nn}\).]

Exercise \(\PageIndex{24}\)

Let \(T : \mathbb{C} \to \mathbb{C}\) be a linear transformation of the real vector space \(\mathbb{C}\) and assume that \(T(a) = a\) for every real number \(a\). Show that the following are equivalent:

1. \(T(zw) = T(z)T(w)\) for all \(z\) and \(w\) in \(\mathbb{C}\).
2. Either \(T = 1_{\mathbb{C}}\) or \(T(z) = \overline{z}\) for each \(z\) in \(\mathbb{C}\) (where \(\overline{z}\) denotes the conjugate).