Any Matrix Represents a Linear Map

For each matrix, state the dimension of the domain and codomain of any map that the matrix represents.

• (a)  $\begin{pmatrix}2&1\\ 3&4\end{pmatrix}$     (b)  $\begin{pmatrix}1&1&-3\\ 2&5&0\end{pmatrix}$     (c)  $\begin{pmatrix}1&3\\ 1&4\\ 1&-1\end{pmatrix}$     (d)  $\begin{pmatrix}0&0&0\\ 0&0&0\end{pmatrix}$     (e)  $\begin{pmatrix}1&-1&4&5\\ 0&0&0&0\end{pmatrix}$

Consider a linear map $f\colon V\to W$ represented with respect to some bases $B,D$ by the matrix. Decide if that map is nonsingular.

• (a)  $\begin{pmatrix}2&1\\ 3&4\end{pmatrix}$     (b)  $\begin{pmatrix}1&1\\ -3&-3\end{pmatrix}$     (c)  $\begin{pmatrix}3&0&0\\ 2&1&0\\ 4&4&4\end{pmatrix}$     (d)  $\begin{pmatrix}2&0&-2\\ 1&1&0\\ 4&1&-4\end{pmatrix}$

Let $h$ be the linear map defined by this matrix on the domain $\mathcal{P}_{1}$ and codomain $\mathbb{R}^{2}$ with respect to the given bases.

What is the image under $h$ of the vector $\vec{v}=2x-1$?

Decide if each vector lies in the range of the map from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$ represented with respect to the standard bases by the matrix.

• (a)  $\begin{pmatrix}1&1&3\\ 0&1&4\end{pmatrix}$, $\begin{pmatrix}1\\ 3\end{pmatrix}$     (b)  $\begin{pmatrix}2&0&3\\ 4&0&6\end{pmatrix}$, $\begin{pmatrix}1\\ 1\end{pmatrix}$

Consider this matrix, representing a transformation of $\mathbb{R}^{2}$ with respect to the bases.

• To what vector in the codomain is the first member of $B$ mapped?

• The second member?

• Where is a general vector from the domain (a vector with components $x$ and $y$) mapped? That is, what transformation of $\mathbb{R}^{2}$ is represented with respect to $B,D$ by this matrix?

Consider a homomorphism $h\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ represented with respect to the standard bases ${\cal E}_{2},{\cal E}_{2}$ by this matrix.

Find the image under $h$ of each vector.

• (a)  $\begin{pmatrix}2\\ 3\end{pmatrix}$     (b)  $\begin{pmatrix}0\\ 1\end{pmatrix}$     (c)  $\begin{pmatrix}-1\\ 1\end{pmatrix}$

What transformation of $F=\text{$\{\hskip 1.00006pta\cos\theta+b\sin\theta\mid a,b\in\mathbb{R}\hskip 1.00006pt\}$}$ is represented with respect to $B=\langle\cos\theta-\sin\theta,\sin\theta\rangle$ and $D=\langle\cos\theta+\sin\theta,\cos\theta\rangle$ by this matrix?

Decide whether $1+2x$ is in the range of the map from $\mathbb{R}^{3}$ to $\mathcal{P}_{2}$ represented with respect to ${\cal E}_{3}$ and $\langle 1,1+x^{2},x\rangle$ by this matrix.

Find the map that this matrix represents with respect to $B,B$.

Example 2.11 gives a matrix that is singular and is therefore associated with maps that are singular. We cannot state the action of the associated map $g$ on domain elements $\vec{v}\in V$, because do not know the domain $V$ or codomain $W$ or the starting and ending bases $B$ and $D$. But we can compute what happens to the representations ${\rm Rep}_{B,D}(\vec{v})$.

• Find the set of column vectors representing the members of the null space of…

Take each matrix to represent $h\colon\mathbb{R}^{m}\to\mathbb{R}^{n}$ with respect to the standard bases. For each (i) state $m$ and $n$. Then set up an augmented matrix with the given matrix on the left and a vector representing a range space element on the right (e.g., if the codomain is $\mathbb{R}^{3}$ then in the right-hand column put the three entries $a$, $b$, and $c$). Perform Gauss-Jordan reduction. Use that to (ii) find $\mathscr{R}(h)$ and $\operatorname{rank}(h)$ (and state whether th…

Use the method from the prior exercise on this matrix.

Verify that the map represented by this matrix is an isomorphism.

This is an alternative proof of Lemma 2.9. Given an $n\!\times\!n$ matrix $H$, fix a domain $V$ and codomain $W$ of appropriate dimension $n$, and bases $B,D$ for those spaces, and consider the map $h$ represented by the matrix.

• Show that $h$ is onto if and only if there is at least one ${\rm Rep}_{B}(\vec{v})$ associated by $H$ with each ${\rm Rep}_{D}(\vec{w})$.

• Show that $h$ is one-to-one if and only if there is at most one ${\rm Rep}_{B}(\vec{v})$ associated by $H$ with each ${\rm Rep}_{D}(\vec{w})$.

• Consider th…

Because the rank of a matrix equals the rank of any map it represents, if one matrix represents two different maps $H={\rm Rep}_{B,D}(h)={\rm Rep}_{\hat{B},\hat{D}}(\hat{h})$ (where $h,\hat{h}\colon V\to W$) then the dimension of the range space of $h$ equals the dimension of the range space of $\hat{h}$. Must these equal-dimensional range spaces actually be the same?

Let $V$ be an $n$-dimensional space with bases $B$ and $D$. Consider a map that sends, for $\vec{v}\in V$, the column vector representing $\vec{v}$ with respect to $B$ to the column vector representing $\vec{v}$ with respect to $D$. Show that map is a linear transformation of $\mathbb{R}^{n}$.

Example 2.3 shows that changing the pair of bases can change the map that a matrix represents, even though the domain and codomain remain the same. Could the map ever not change? Is there a matrix $H$, vector spaces $V$ and $W$, and associated pairs of bases $B_{1},D_{1}$ and $B_{2},D_{2}$ (with $B_{1}\neq B_{2}$ or $D_{1}\neq D_{2}$ or both) such that the map represented by $H$ with respect to $B_{1},D_{1}$ equals the map represented by $H$ with respect to $B_{2},D_{2}$?

A square matrix is a diagonal matrix if it is all zeroes except possibly for the entries on its upper-left to lower-right diagonal — its $1,1$ entry, its $2,2$ entry, etc. Show that a linear map is an isomorphism if there are bases such that, with respect to those bases, the map is represented by a diagonal matrix with no zeroes on the diagonal.

Describe geometrically the action on $\mathbb{R}^{2}$ of the map represented with respect to the standard bases ${\cal E}_{2},{\cal E}_{2}$ by this matrix.

Do the same for these.

The fact that for any linear map the rank plus the nullity equals the dimension of the domain shows that a necessary condition for the existence of a homomorphism between two spaces, onto the second space, is that there be no gain in dimension. That is, where $h\colon V\to W$ is onto, the dimension of $W$ must be less than or equal to the dimension of $V$.

• Show that this (strong) converse holds: no gain in dimensi…

Let $V$ be an $n$-dimensional space and suppose that $\vec{x}\in\mathbb{R}^{n}$. Fix a basis $B$ for $V$ and consider the map $h_{\vec{x}}\colon V\to\mathbb{R}$ given $\vec{v}\mapsto\vec{x}\mathbin{\smash{\bullet}}{\rm Rep}_{B}(\vec{v})$ by the dot product.

• Show that this map is linear.

• Show that for any linear map $g\colon V\to\mathbb{R}$ there is an $\vec{x}\in\mathbb{R}^{n}$ such that $g=h_{\vec{x}}$.

• In the prior item we fixed the basis and varied the $\vec{x}$ to get all possible linear maps. Can we get all possible linear maps by fixing an $\vec{x}$ and varying the basis?

Let $V,W,X$ be vector spaces with bases $B,C,D$.

• Suppose that $h\colon V\to W$ is represented with respect to $B,C$ by the matrix $H$. Give the matrix representing the scalar multiple $rh$ (where $r\in\mathbb{R}$) with respect to $B,C$ by expressing it in terms of $H$.

• Suppose that $h,g\colon V\to W$ are represented with respect to $B,C$ by $H$ and $G$. Give the matrix representing $h+g$ with respect to $B,C$ by expressing it in terms of $H$ and $G$.

• Suppose that $h\colon V\to W$ is represented wit…