Linear Transformations

Let $\mathcal{U}$ represent a subspace of a linear space $\mathcal{V}$, and let $S$ represent a linear transformation from $\mathcal{U}$ into a linear space $\mathcal{W}$. Show that there exists a linear transformation $T$ from $\mathcal{V}$ into $\mathcal{W}$ such that $S$ is the restriction of $T$ to $\mathcal{U}$.

Let $T$ represent a 1-1 linear transformation from a linear space $\mathcal{V}$ into a linear space $\mathcal{W}$. And, write $\mathbf{U} \cdot \mathbf{Y}$ for the inner product of arbitrary matrices $\mathbf{U}$ and $\mathbf{Y}$ in $\mathcal{W}$. Further, define $\mathbf{X} * \mathbf{Z} = T(\mathbf{X}) \cdot T(\mathbf{Z})$ for all matrices $\mathbf{X}$ and $\mathbf{Z}$ in $\mathcal{V}$. Show that the ``*-operation'' satisfies the four properties required of an inner product for $\mathcal{V}$.

Let $T$ represent a linear transformation from a linear space $\mathcal{V}$ into a linear space $\mathcal{W}$, and let $\mathcal{U}$ represent any subspace of $\mathcal{V}$ such that $\mathcal{U}$ and $N(T)$ are essentially disjoint. Further, let $\{ \mathbf{X}_{1}, \ldots , \mathbf{X}_{r}\}$ represent a linearly independent set of $r$ matrices in $\mathcal{U}$.

Let $T$ and $S$ represent linear transformations from a linear space $\mathcal{V}$ into a linear space $\mathcal{W}$, and let $k$ represent an arbitrary scalar.

Let $S$ represent a linear transformation from a linear space $\mathcal{U}$ into a linear space $\mathcal{V}$, and let $T$ represent a linear transformation from $\mathcal{V}$ into a linear space $\mathcal{W}$. Show that the transformation $TS$ is linear.

Let $T$ represent a linear transformation from a linear space $\mathcal{V}$ into a linear space $\mathcal{W}$, and let $R$ represent a linear transformation from a linear space $\mathcal{U}$ into $\mathcal{W}$. Show that if $T(\mathcal{V}) \subset R(\mathcal{U})$, then there exists a linear transformation $S$ from $\mathcal{V}$ into $\mathcal{U}$ such that $T = RS$.

Let $T$ represent a transformation from a linear space $\mathcal{V}$ into a linear space $\mathcal{W}$, and let $S$ and $R$ represent transformations from $\mathcal{W}$ into $\mathcal{V}$. And, suppose that $RT = I$ (where the identity transformation $I$ is from $\mathcal{V}$ onto $\mathcal{V}$) and that $TS = I$ (where the identity transformation $I$ is from $\mathcal{W}$ onto $\mathcal{W}$).

Let $T$ represent an invertible transformation from a linear space $\mathcal{V}$ into a linear space $\mathcal{W}$, let $S$ represent an invertible transformation from a linear space $\mathcal{U}$ into $\mathcal{V}$, and let $k$ represent an arbitrary scalar. Using the results of Exercise 7 (or otherwise), show that

Let $T$ represent a linear transformation from an $n$-dimensional linear space $\mathcal{V}$ into an $m$-dimensional linear space $\mathcal{W}$. And, write $\mathbf{U} \cdot \mathbf{Y}$ for the inner product of arbitrary matrices $\mathbf{U}$ and $\mathbf{Y}$ in $\mathcal{W}$. Further, let $B$ represent a set of matrices $\mathbf{V}_{1}, \ldots , \mathbf{V}_{n}$ (in $\mathcal{V}$) that form a basis for $\mathcal{V}$, and let $C$ represent a set of matrices $\mathbf{W}_{1}, \ldots , \mathbf{W}_{m}$ (in $\mathcal{W}$) that form an orthonormal basis for $\mathcal{W}$. Show that the matrix representation of $T$ with respect to $B$ and $C$ is the $m \times n$ matrix whose $ij$th element is $T(\mathbf{V}_{j}) \cdot \mathbf{W}_{i}$.

Let $T$ represent the linear transformation from $\mathbb {R}^{m \times n}$ into $\mathbb {R}^{n \times m}$ defined by $T(\mathbf{X}) = \mathbf{X}'$. And, let $C$ represent the natural basis for $\mathbb {R}^{m \times n}$, comprising the $mn$ matrices $\mathbf{U}_{11}, \mathbf{U}_{21}, \ldots , \mathbf{U}_{m1}, \ldots , \mathbf{U}_{1n}, \mathbf{U}_{2n}, \ldots , \mathbf{U}_{mn}$, where (for $i = 1, \ldots , m$ and $j = 1, \ldots , n$) $\mathbf{U}_{ij}$ is the $m \times n$ matrix whose $ij$th element equals 1 and whose remaining $mn-1$ elements equal 0; and similarly let $D$ represent the natural basis for $\mathbb {R}^{n \times m}$. Show that the matrix representation for $T$ with respect to the bases $C$ and $D$ is the vec-permutation matrix $\mathbf{K}_{mn}$.

Let $\mathcal{W}$ represent the linear space of all $p \times p$ symmetric matrices, and let $T$ represent a linear transformation from $\mathbb {R}^{m \times n}$ into $\mathcal{W}$. Further, let $B$ represent the natural basis for $\mathbb {R}^{m \times n}$, comprising the $mn$ matrices $\mathbf{U}_{11}, \mathbf{U}_{21}, \ldots , \mathbf{U}_{m1}, \ldots , \mathbf{U}_{1n}, \ldots , \mathbf{U}_{mn}$, where (for $i = 1, \ldots , m$ and $j = 1, \ldots , n$) $\mathbf{U}_{ij}$ is the $m \times n$ matrix whose $ij$th element equals 1 and whose remaining $mn-1$ elements equal 0. And, let $C$ represent the usual basis for $\mathcal{W}$.

Let $\mathcal{V}$ represent an $n$-dimensional linear space. Further, let $B = \{ \mathbf{V}_{1}, \mathbf{V}_{2}, \ldots , \mathbf{V}_{n}\}$ represent a basis for $\mathcal{V}$, and let $\mathbf{A}$ represent an $n \times n$ nonsingular matrix. And, for $j = 1, \ldots , n$, let

where (for $i = 1, \ldots , n$) $f_{ij}$ is the $ij$th element of $\mathbf{A}^{-1}$. Show that the set $C$ comprising the matrices $\mathbf{W}_{1}, \mathbf{W}_{2}, \ldots , \mathbf{W}_{n}$ is a basis for $\mathcal{V}$ and that $\mathbf{A}$ is the matrix representation of the identity transformation $I$ (from $\mathcal{V}$ onto $\mathcal{V}$) with respect to $B$ and $C$.

Let $T$ represent the linear transformation from $\mathbb {R}^{4 \times 1}$ into $\mathbb {R}^{3 \times 1}$ defined by

where $\mathbf{x} = (x_{1}, x_{2}, x_{3}, x_{4})'$. Further, let $B$ represent the natural basis for $\mathbb {R}^{4 \times 1}$ (comprising the columns of $\mathbf{I}_{4}$), and let $E$ represent the basis (for $\mathbb {R}^{4 \times 1}$) comprising the four vectors $(1,-1,0,-1)'$, $(0,0,1,1)'$, $(0,0,0,1)'$, and $(1,1,0,0)'$. And, let $C$ represent the natural basis for $\mathbb {R}^{3 \times 1}$ (comprising the columns of $\mathbf{I}_{3}$), and $F$ represent the basis (for $\mathbb {R}^{3 \times 1}$) comprising the three vectors $(1,0,1)'$, $(1,1,0)'$, and $(-1,0,0)'$.

Let $T$ represent a linear transformation of rank $k$ (where $k > 0$) from an $n$-dimensional linear space $\mathcal{V}$ into an $m$-dimensional linear space $\mathcal{W}$. Show that there exists a basis $E$ for $\mathcal{V}$ and a basis $F$ for $\mathcal{W}$ such that the matrix representation of $T$ with respect to $E$ and $F$ is of the form $\begin{pmatrix} \mathbf{I}_{k} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{pmatrix}$.

Let $T$ represent a linear transformation from an $n$-dimensional linear space $\mathcal{V}$ into an $m$-dimensional linear space $\mathcal{W}$, and let $\mathbf{A}$ represent the matrix representation of $T$ with respect to bases $B$ and $C$ (for $\mathcal{V}$ and $\mathcal{W}$, respectively). Use the result that an $n \times 1$ vector $\mathbf{x}$ is in $N(\mathbf{A})$ if and only if the corresponding matrix $L_{B}(\mathbf{x})$ is in $N(T)$ [or equivalently that a matrix $\mathbf{X}$ (in $\mathcal{V}$) is in $N(T)$ if and only if the corresponding vector $L_{B}^{-1}(\mathbf{X})$ is in $N(\mathbf{A})$] to devise a ``direct'' proof that $\dim [N(T)] = \dim [N(\mathbf{A})]$ (as opposed to deriving this equality as a corollary of the result that $\operatorname {rk} T = \operatorname {rk}\left(\mathbf{A}\right)$).

Let $T$ represent a linear transformation from an $n$-dimensional linear space $\mathcal{V}$ into $\mathcal{V}$.

Let $\mathcal{V}$, $\mathcal{W}$, and $\mathcal{U}$ represent linear spaces.

Let $S$ represent the dual transformation of a linear transformation $T$ from an $n$-dimensional linear space $\mathcal{V}$ into an $m$-dimensional linear space $\mathcal{W}$. And, let $\mathbf{A} = \{ a_{ij}\}$ represent the matrix representation of $T$ with respect to orthonormal bases $C$ and $D$, and $\mathbf{B} = \{ b_{ij}\}$ represent the matrix representation of $S$ with respect to $D$ and $C$. Using the result of Exercise 9 (or otherwise), show that $\mathbf{B} = \mathbf{A}'$.

Let $\mathbf{A}$ represent an $m \times n$ matrix, let $\mathbf{V}$ represent an $n \times n$ symmetric positive definite matrix, and let $\mathbf{W}$ represent an $m \times m$ symmetric positive definite matrix. And let $S$ represent the dual transformation of the linear transformation $T$ from $\mathbb {R}^{n \times 1}$ into $\mathbb {R}^{m \times 1}$ defined by $T(\mathbf{x}) = \mathbf{A}\mathbf{x}$ (where $\mathbf{x}$ is an arbitrary $n \times 1$ vector). Taking the inner product of arbitrary vectors $\mathbf{x}$ and $\mathbf{z}$ in $\mathbb {R}^{n \times 1}$ to be $\mathbf{x}'\mathbf{V}\mathbf{z}$ and the inner product of arbitrary vectors $\mathbf{u}$ and $\mathbf{y}$ in $\mathbb {R}^{m \times 1}$ to be $\mathbf{u}'\mathbf{W}\mathbf{y}$, obtain a formula for $S(\mathbf{y})$ that generalizes the formula $S(\mathbf{y}) = \mathbf{A}'\mathbf{y}$ ($\mathbf{y} \in \mathbb {R}^{m \times 1}$) obtained in the special case of the usual inner products (i.e., in the special case where $\mathbf{V} = \mathbf{I}_{n}$ and $\mathbf{W} = \mathbf{I}_{m}$).

Let $S$ represent the dual transformation of a linear transformation $T$ from a linear space $\mathcal{V}$ into a linear space $\mathcal{W}$.