Math Problems Vault

Problem 4.7.1

Determine which of the following functions are linear operators on $\mathbb {R}^{2}$ .

?

(a)

$\mathbf{T}(x,y) = (x, 1+y)$ ;

(b)

$\mathbf{T}(x,y) = (y, x)$ ;

(c)

$\mathbf{T}(x,y) = (0, xy)$ ;

(d)

$\mathbf{T}(x,y) = (x^2, y^2)$ ;

(e)

$\mathbf{T}(x,y) = (x, \sin y)$ ;

(f)

$\mathbf{T}(x,y) = (x+y, x-y)$ .

Problem 4.7.2

(a)(b)(c)(d)

For $\mathbf{A} \in \mathbb {R}^{n \times n}$ , determine which of the following functions are linear transformations.

?

(a)

$\mathbf{T}(\mathbf{X}_{n \times n}) = \mathbf{A}\mathbf{X} - \mathbf{X}\mathbf{A}$ ;

(b)

$\mathbf{T}(\overrightarrow {x}_{n \times 1}) = \mathbf{A}\overrightarrow {x} + \overrightarrow {b}$ for $\overrightarrow {b} \neq \overrightarrow {0}$ ;

(c)

$\mathbf{T}(\mathbf{A}) = \mathbf{A}^T$ ;

(d)

$\mathbf{T}(\mathbf{X}_{n \times n}) = (\mathbf{X} + \mathbf{X}^T)/2$ .

Problem 4.7.3

Explain why $\mathbf{T}(\overrightarrow {0}) = \overrightarrow {0}$ for every linear transformation $\mathbf{T}$ .

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Problem 4.7.4

(a)(b)

Determine which of the following mappings are linear operators on $\mathcal{P}_n$ , the vector space of polynomials of degree $n$ or less.

?

(a)

$\mathbf{T} = \xi_k \mathbf{D}^k + \xi_{k-1} \mathbf{D}^{k-1} + \cdots + \xi_1 \mathbf{D} + \xi_0 \mathbf{I}$ , where $\mathbf{D}^k$ is the $k^{th}$ -order differentiation operator (i.e., $\mathbf{D}^k p(t) = d^k p/dt^k$ ).

(b)

$\mathbf{T}\left(p(t)\right) = t^n p'(0) + t$ .

Problem 4.7.5

(a)(b)

Let $\overrightarrow {v}$ be a fixed vector in $\mathbb {R}^{n \times 1}$ and let $\mathbf{T} : \mathbb {R}^{n \times 1} \rightarrow \mathbb {R}$ be the mapping defined by $\mathbf{T}(\overrightarrow {x}) = \overrightarrow {v}^T \overrightarrow {x}$ (i.e., the standard inner product).

?

(a)

Is $\mathbf{T}$ a linear operator?

(b)

Is $\mathbf{T}$ a linear transformation?

Problem 4.7.6

For the operator $\mathbf{T} : \mathbb {R}^{2} \rightarrow \mathbb {R}^{2}$ defined by $\mathbf{T}(x,y) = (x+y, -2x+4y)$ , determine $[\mathbf{T}]_{\mathcal{B}}$ , where $\mathcal{B}$ is the basis $\mathcal{B} = \left\{ \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 2 \end{pmatrix}\right\}$ .

?

Problem 4.7.7

(a)(b)

Let $\mathbf{T} : \mathbb {R}^{2} \rightarrow \mathbb {R}^{3}$ be the linear transformation defined by $\mathbf{T}(x,y) = (x+3y, 0, 2x-4y)$ .

?

(a)

Determine $[\mathbf{T}]_{\mathcal{S}\mathcal{S}'}$ , where $\mathcal{S}$ and $\mathcal{S}'$ are the standard bases for $\mathbb {R}^{2}$ and $\mathbb {R}^{3}$ , respectively.

(b)

Determine $[\mathbf{T}]_{\mathcal{S}\mathcal{S}''}$ , where $\mathcal{S}''$ is the basis for $\mathbb {R}^{3}$ obtained by permuting the standard basis according to $\mathcal{S}'' = \left\{ \overrightarrow {e}_3, \overrightarrow {e}_2, \overrightarrow {e}_1\right\}$ .

Problem 4.7.8

(a)(b)

Let $\mathbf{T}$ be the operator on $\mathbb {R}^{3}$ defined by $\mathbf{T}(x,y,z) = (x-y, y-x, x-z)$ and consider the vector

\overrightarrow {v} = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} \quad \text{and the basis} \quad \mathcal{B} = \left\{ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}\right\} .

?

(a)

Determine $[\mathbf{T}]_{\mathcal{B}}$ and $[\overrightarrow {v}]_{\mathcal{B}}$ .

(b)

Compute $[\mathbf{T}(\overrightarrow {v})]_{\mathcal{B}}$ , and then verify that $[\mathbf{T}]_{\mathcal{B}}[\overrightarrow {v}]_{\mathcal{B}} = [\mathbf{T}(\overrightarrow {v})]_{\mathcal{B}}$ .

Problem 4.7.9

For $\mathbf{A} \in \mathbb {R}^{n \times n}$ , let $\mathbf{T}$ be the linear operator on $\mathbb {R}^{n \times 1}$ defined by $\mathbf{T}(\overrightarrow {x}) = \mathbf{A}\overrightarrow {x}$ . That is, $\mathbf{T}$ is the operator defined by matrix multiplication. With respect to the standard basis $\mathcal{S}$ , show that $[\mathbf{T}]_{\mathcal{S}} = \mathbf{A}$ .

Formula (4.7.4): if $\mathcal{B} = \left\{ \overrightarrow {u}_i\right\}_{i=1}^n$ and $\mathcal{B}' = \left\{ \overrightarrow {v}_i\right\}_{i=1}^m$ are bases and $\mathbf{T}(\overrightarrow {u}_j) = \sum_i \alpha_{ij} \overrightarrow {v}_i$ , then $[\mathbf{T}]_{\mathcal{B}\mathcal{B}'} = (\alpha_{ij})$ is the coordinate matrix representation of $\mathbf{T}$ , satisfying $[\mathbf{T}(\overrightarrow {u})]_{\mathcal{B}'} = [\mathbf{T}]_{\mathcal{B}\mathcal{B}'}[\overrightarrow {u}]_{\mathcal{B}}$ for every $\overrightarrow {u}$ .

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Problem 4.7.10

If $\mathbf{T}$ is a linear operator on a space $\mathcal{V}$ with basis $\mathcal{B}$ , explain why $[\mathbf{T}^k]_{\mathcal{B}} = [\mathbf{T}]_{\mathcal{B}}^k$ for all nonnegative integers $k$ .

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Problem 4.7.11

(a)(b)

Let $\mathbf{P}$ be the projector that maps each point $\overrightarrow {v} \in \mathbb {R}^{2}$ to its orthogonal projection on the line $y = x$ .

Figure 4.7.4

?

(a)

Determine the coordinate matrix of $\mathbf{P}$ with respect to the standard basis.

(b)

Determine the orthogonal projection of $\overrightarrow {v} = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$ onto the line $y = x$ .

Problem 4.7.12

(a)(b)

For the standard basis $\mathcal{S} = \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\right\}$ of $\mathbb {R}^{2 \times 2}$ , determine the matrix representation $[\mathbf{T}]_{\mathcal{S}}$ for each of the following linear operators on $\mathbb {R}^{2 \times 2}$ , and then verify $[\mathbf{T}(\mathbf{U})]_{\mathcal{S}} = [\mathbf{T}]_{\mathcal{S}}[\mathbf{U}]_{\mathcal{S}}$ for $\mathbf{U} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ .

?

(a)

$\mathbf{T}(\mathbf{X}_{2 \times 2}) = \dfrac {\mathbf{X} + \mathbf{X}^T}{2}$ .

(b)

$\mathbf{T}(\mathbf{X}_{2 \times 2}) = \mathbf{A}\mathbf{X} - \mathbf{X}\mathbf{A}$ , where $\mathbf{A} = \begin{pmatrix} 1 & 1 \\ -1 & -1 \end{pmatrix}$ .

Problem 4.7.13

For $\mathcal{P}_2$ and $\mathcal{P}_3$ (the spaces of polynomials of degrees less than or equal to two and three, respectively), let $\mathbf{S} : \mathcal{P}_2 \rightarrow \mathcal{P}_3$ be the linear transformation defined by $\mathbf{S}(p) = \int_0^t p(x)\, dx$ . Determine $[\mathbf{S}]_{\mathcal{B}\mathcal{B}'}$ , where $\mathcal{B} = \left\{ 1, t, t^2\right\}$ and $\mathcal{B}' = \left\{ 1, t, t^2, t^3\right\}$ .

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Problem 4.7.14

(a)(b)

Let $\mathbf{Q}$ be the linear operator on $\mathbb {R}^{2}$ that rotates each point counterclockwise through an angle $\theta$ , and let $\mathbf{R}$ be the linear operator on $\mathbb {R}^{2}$ that reflects each point about the $x$ -axis.

?

(a)

Determine the matrix of the composition $[\mathbf{R}\mathbf{Q}]_{\mathcal{S}}$ relative to the standard basis $\mathcal{S}$ .

(b)

Relative to the standard basis, determine the matrix of the linear operator that rotates each point in $\mathbb {R}^{2}$ counterclockwise through an angle $2\theta$ .

Problem 4.7.15

(a)(b)

Let $\mathbf{P} : \mathcal{U} \rightarrow \mathcal{V}$ and $\mathbf{Q} : \mathcal{U} \rightarrow \mathcal{V}$ be two linear transformations, and let $\mathcal{B}$ and $\mathcal{B}'$ be arbitrary bases for $\mathcal{U}$ and $\mathcal{V}$ , respectively.

?

(a)

Provide the details to explain why $[\mathbf{P} + \mathbf{Q}]_{\mathcal{B}\mathcal{B}'} = [\mathbf{P}]_{\mathcal{B}\mathcal{B}'} + [\mathbf{Q}]_{\mathcal{B}\mathcal{B}'}$ .

(b)

Provide the details to explain why $[\alpha \mathbf{P}]_{\mathcal{B}\mathcal{B}'} = \alpha [\mathbf{P}]_{\mathcal{B}\mathcal{B}'}$ , where $\alpha$ is an arbitrary scalar.

Problem 4.7.16

(a)(b)(c)

Let $\mathbf{I}$ be the identity operator on an $n$ -dimensional space $\mathcal{V}$ .

?

(a)

Explain why

[\mathbf{I}]_{\mathcal{B}} = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}

regardless of the choice of basis $\mathcal{B}$ .

(b)

Let $\mathcal{B} = \left\{ \overrightarrow {x}_i\right\}_{i=1}^n$ and $\mathcal{B}' = \left\{ \overrightarrow {y}_i\right\}_{i=1}^n$ be two different bases for $\mathcal{V}$ , and let $\mathbf{T}$ be the linear operator on $\mathcal{V}$ that maps vectors from $\mathcal{B}'$ to vectors in $\mathcal{B}$ according to the rule $\mathbf{T}(\overrightarrow {y}_i) = \overrightarrow {x}_i$ for $i = 1, 2, \ldots , n$ . Explain why

[\mathbf{I}]_{\mathcal{B}\mathcal{B}'} = [\mathbf{T}]_{\mathcal{B}} = [\mathbf{T}]_{\mathcal{B}'} = \left( [\overrightarrow {x}_1]_{\mathcal{B}'} \; \middle | \; [\overrightarrow {x}_2]_{\mathcal{B}'} \; \middle | \; \cdots \; \middle | \; [\overrightarrow {x}_n]_{\mathcal{B}'} \right).

(c)

When $\mathcal{V} = \mathbb {R}^{3}$ , determine $[\mathbf{I}]_{\mathcal{B}\mathcal{B}'}$ for

\mathcal{B} = \left\{ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\right\} , \qquad \mathcal{B}' = \left\{ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\right\} .

Problem 4.7.17

(a)(b)

Let $\mathbf{T} : \mathbb {R}^{3} \rightarrow \mathbb {R}^{3}$ be the linear operator defined by

\mathbf{T}(x,y,z) = (2x-y, \; -x+2y-z, \; z-y).

?

(a)

Determine $\mathbf{T}^{-1}(x,y,z)$ .

(b)

Determine $[\mathbf{T}^{-1}]_{\mathcal{S}}$ , where $\mathcal{S}$ is the standard basis for $\mathbb {R}^{3}$ .

Problem 4.7.18

Let $\mathbf{T}$ be a linear operator on an $n$ -dimensional space $\mathcal{V}$ . Show that the following statements are equivalent.

$\mathbf{T}^{-1}$ exists.
$\mathbf{T}$ is a one-to-one mapping (i.e., $\mathbf{T}(\overrightarrow {x}) = \mathbf{T}(\overrightarrow {y}) \implies \overrightarrow {x} = \overrightarrow {y}$ ).
$N(\mathbf{T}) = \left\{ \overrightarrow {0}\right\}$ .
$\mathbf{T}$ is an onto mapping (i.e., for each $\overrightarrow {v} \in \mathcal{V}$ , there is an $\overrightarrow {x} \in \mathcal{V}$ such that $\mathbf{T}(\overrightarrow {x}) = \overrightarrow {v}$ ).

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Problem 4.7.19

(a)(b)

Let $\mathcal{V}$ be an $n$ -dimensional space with a basis $\mathcal{B} = \left\{ \overrightarrow {u}_i\right\}_{i=1}^n$ .

?

(a)

Prove that a set of vectors $\left\{ \overrightarrow {x}_1, \overrightarrow {x}_2, \ldots , \overrightarrow {x}_r\right\} \subseteq \mathcal{V}$ is linearly independent if and only if the set of coordinate vectors

\left\{ [\overrightarrow {x}_1]_{\mathcal{B}}, \; [\overrightarrow {x}_2]_{\mathcal{B}}, \; \ldots , \; [\overrightarrow {x}_r]_{\mathcal{B}}\right\} \subseteq \mathbb {R}^{n \times 1}

is a linearly independent set.

(b)

If $\mathbf{T}$ is a linear operator on $\mathcal{V}$ , then the range of $\mathbf{T}$ is the set

R(\mathbf{T}) = \left\{ \mathbf{T}(\overrightarrow {x}) \mid \overrightarrow {x} \in \mathcal{V}\right\} .

Suppose that the basic columns of $[\mathbf{T}]_{\mathcal{B}}$ occur in positions $b_1, b_2, \ldots , b_r$ . Explain why $\left\{ \mathbf{T}(\overrightarrow {u}_{b_1}), \mathbf{T}(\overrightarrow {u}_{b_2}), \ldots , \mathbf{T}(\overrightarrow {u}_{b_r})\right\}$ is a basis for $R(\mathbf{T})$ .