Representing Linear Maps with Matrices

Multiply the matrix

by each vector, or state “not defined.”

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

Multiply this matrix by each vector or state “not defined.”

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

Perform, if possible, each matrix-vector multiplication.

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

This matrix equation expresses a linear system. Solve it.

For a homomorphism from $\mathcal{P}_{2}$ to $\mathcal{P}_{3}$ that sends

where does $1-3x+2x^{2}$ go?

Let $h\colon\mathbb{R}^{2}\to\mathcal{M}_{2\!\times\!2}$ be the linear transformation with this action.

What is its effect on the general vector with entries $x$ and $y$?

Assume that $h\colon\mathbb{R}^{2}\to\mathbb{R}^{3}$ is determined by this action.

Using the standard bases, find

• the matrix representing this map;

• a general formula for $h(\vec{v})$.

Represent the homomorphism $h\colon\mathbb{R}^{3}\to\mathbb{R}^{2}$ given by this formula and with respect to these bases.

Let $d/dx\colon\mathcal{P}_{3}\to\mathcal{P}_{3}$ be the derivative transformation.

• Represent $d/dx$ with respect to $B,B$ where $B=\langle 1,x,x^{2},x^{3}\rangle$.

• Represent $d/dx$ with respect to $B,D$ where $D=\langle 1,2x,3x^{2},4x^{3}\rangle$.

Represent each linear map with respect to each pair of bases.

• $d/dx\colon\mathcal{P}_{n}\to\mathcal{P}_{n}$ with respect to $B,B$ where $B=\langle 1,x,\dots,x^{n}\rangle$, given by

• $\int\colon\mathcal{P}_{n}\to\mathcal{P}_{n+1}$ with respect to $B_{n},B_{n+1}$ where $B_{i}=\langle 1,x,\dots,x^{i}\rangle$, given by

• $\int^{1}_{0}\colon\mathcal{P}_{n}\to\mathbb{R}$ with respect to $B,{\cal E}_{1}$ where $B=\langle 1,x,\dots,x^{n}\rangle$ and ${\cal E}_{1}=\langle 1\rangle$, given by

• $\text{eval}_{3}\colon\mathcal{P}_{n}\to\mathbb{R}$ with respect to $B,{\cal E}_{1}$ where $B=\langle 1,x,\dots,x^{n}\rangle$ and ${\cal E}_{1}=\langle 1\rangle$, given by

• $\text{slide}_{-1}\colon\mathcal{P}_{n}\to\mathcal{P}_{n}$ with respect to $B,B$ where $B=\langle 1,x,\ldots,x^{n}\rangle$, given by

Represent the identity map on any nontrivial space with respect to $B,B$, where $B$ is any basis.

Represent, with respect to the natural basis, the transpose transformation on the space $\mathcal{M}_{2\!\times\!2}$ of $2\!\times\!2$ matrices.

Assume that $B=\langle\vec{\beta}_{1},\vec{\beta}_{2},\vec{\beta}_{3},\vec{\beta}_{4}\rangle$ is a basis for a vector space. Represent with respect to $B,B$ the transformation that is determined by each.

• $\vec{\beta}_{1}\mapsto\vec{\beta}_{2}$, $\vec{\beta}_{2}\mapsto\vec{\beta}_{3}$, $\vec{\beta}_{3}\mapsto\vec{\beta}_{4}$, $\vec{\beta}_{4}\mapsto\vec{0}$

• $\vec{\beta}_{1}\mapsto\vec{\beta}_{2}$, $\vec{\beta}_{2}\mapsto\vec{0}$, $\vec{\beta}_{3}\mapsto\vec{\beta}_{4}$, $\vec{\beta}_{4}\mapsto\vec{0}$

• $\vec{\beta}_{1}\mapsto\vec{\beta}_{2}$, $\vec{\beta}_{2}\mapsto\vec{\beta}_{3}$, $\vec{\beta}_{3}\mapsto\vec{0}$, $\vec{\beta}_{4}\mapsto\vec{0}$

Consider a linear transformation of $\mathbb{R}^{2}$ determined by these two.

• Represent this transformation with respect to the standard bases.

• Where does the transformation send this vector?

• Represent this transformation with respect to these bases.

• Using $B$ from the prior item, represent the transformation with respect to $B,B$.

Suppose that $h\colon V\to W$ is one-to-one so that by Theorem 2.20, for any basis $B=\langle\vec{\beta}_{1},\dots,\vec{\beta}_{n}\rangle\subset V$ the image $h(B)=\langle h(\vec{\beta}_{1}),\dots,h(\vec{\beta}_{n})\rangle$ is a basis for $h(V)$. (If $h$ is onto then $h(V)=W$.)

• Represent the map $h$ with respect to $\langle B,h(B)\rangle$.

• For a member $\vec{v}$ of the domain, where the representation of $\vec{v}$ has components $c_{1}$, …, $c_{n}$, represent the image vector $h(\vec{v})$ with respect to the image basis $h(B)$.

Give a formula for the product of a matrix and $\vec{e}_{i}$, the column vector that is all zeroes except for a single one in the $i$-th position.

For each vector space of functions of one real variable, represent the derivative transformation with respect to $B,B$.

• $\{\hskip 1.00006pta\cos x+b\sin x\mid a,b\in\mathbb{R}\hskip 1.00006pt\}$, $B=\langle\cos x,\sin x\rangle$

• $\{\hskip 1.00006ptae^{x}+be^{2x}\mid a,b\in\mathbb{R}\hskip 1.00006pt\}$, $B=\langle e^{x},e^{2x}\rangle$

• $\{\hskip 1.00006pta+bx+ce^{x}+dxe^{x}\mid a,b,c,d\in\mathbb{R}\hskip 1.00006pt\}$, $B=\langle 1,x,e^{x},xe^{x}\rangle$

Find the range of the linear transformation of $\mathbb{R}^{2}$ represented with respect to the standard bases by each matrix.

• (a)  $\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$     (b)  $\begin{pmatrix}0&0\\ 3&2\end{pmatrix}$     (c)  a matrix of the form $\begin{pmatrix}a&b\\ 2a&2b\end{pmatrix}$

Can one matrix represent two different linear maps? That is, can ${\rm Rep}_{B,D}(h)={\rm Rep}_{\hat{B},\hat{D}}(\hat{h})$?

Prove Theorem 1.5.

(Schur’s Triangularization Lemma)

• Let $U$ be a subspace of $V$ and fix bases $B_{U}\subseteq B_{V}$. What is the relationship between the representation of a vector from $U$ with respect to $B_{U}$ and the representation of that vector (viewed as a member of $V$) with respect to $B_{V}$?

• What about maps?

• Fix a basis $B=\langle\vec{\beta}_{1},\dots,\vec{\beta}_{n}\rangle$ for $V$ and observe that the spans

  form a strictly increasing chain of subspaces. Show that for any linear map $h\colon V\to W$ the…