# Linear Exam 2

## Instructions

There are 15 problems on this exam, and you only need to turn in 6 of them.  You may choose any 6, and you may consult your notes, our textbook, and me while working on them.  You are not allowed to use any other resources or discuss your work with any other people.  The exam is due on Tuesday, April 5th.

Note: each question is worth 20 points.

Remember to write clearly and to show your work.  Enjoy!

## The Actual Exam

Question 1.  Prove that if $h \in \mathcal{L}(V,W)$ is an isomorphism, then $h^{-1} \in \mathcal{L}(W, V)$.

Question 2.  Let $\mathcal{B} = \{\vec{v}_1, \vec{v}_2, \cdots, \vec{v}_n\}$ be a basis for $V = \mathbb{R}^n$, and let $\mathcal{E}_n$ denote the standard basis for this space.

(a) Explain why the matrix representation of the identity map, $\text{id}_{\mathbb{R}^n}:\mathbb{R}^n \to \mathbb{R}^n$, has as its matrix representation (relative to the bases $\mathcal{B}$ for the domain and $\mathcal{E}_n$ for the codomain)

$\text{Rep}_{\mathcal{B},\mathcal{E}_n}\,\left(\text{id}_{\mathbb{R}^n}\right) = \left[\begin{array}{cccc} | & | & & | \\ \vec{v}_1 & \vec{v}_2 & \cdots & \vec{v}_n \\ | & | & & | \end{array}\right]$

(b) Let $h \in \mathcal{L}(\mathbb{R}^n, \mathbb{R}^n)$ and suppose $\mathcal{B} = \{\vec{v}_1, \vec{v}_2, \cdots, \vec{v}_n\}$ is a basis for $\mathbb{R}^n$.  Moreover, also suppose that each vector in this basis is an eigenvector for the homomorphism $h$, i.e. suppose that

$h(\vec{v}_i) = \lambda_i\,\vec{v}_i$

where the scalars $\lambda_i \in \mathbb{R}$.  Explain / prove that $\text{Rep}_{\mathcal{B},\mathcal{B}}\,(h)$ is the diagonal matrix

$\text{Rep}_{\mathcal{B}, \mathcal{B}}\,(h) = \left[ \begin{array}{cccc} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{array}\right]$

Question 3.  Let $T:\mathcal{P}_2 \to \mathcal{P}_2$ be the function

$T(p(x)) = x^2p''(x)$.

(a) Prove that $T$ is linear.

(b) Prove that $T$ is one-to-one, or find two inputs in $\mathcal{P}_2$ that have the same output.

(c ) Prove that $T$ is onto, or find an element in the codomain that is not in the range space of $T$.

(d) Compute the matrix representation of $T$ relative to the bases $\mathcal{B} = \{1, 2x, 3x^2\}$ for the domain and $\mathcal{D} = \{2, 1+x, 2+x+x^2\}$ for the co-domain.

Question 4.  Let $A$ be an $m \times n$ matrix.

(a) The column space of $A$ is the subspace of $\mathbb{R}^m$ that is spanned by the columns of $A$ (where each column is viewed as a column vector in $\mathbb{R}^m$).

Prove that the column space of $A$ equals the range space of $A$.  Note that the range space of a matrix is the set of outputs

$\mathcal{R}(A) = \{A\vec{x} : \vec{x} \in \mathbb{R}^n\}$

where $A$ multiplies against vectors $\vec{x} \in \mathbb{R}^n$ via matrix multiplication.

(b) The row space of $A$ is the subspace of $\mathbb{R}^n$ that is spanned by the rows of $A$ (where each row is viewed as a vector in $\mathbb{R}^n$).  Explain / prove that the dimension of the row space always equals the dimension of the column space.

Question 5.  Suppose $h:V\to W$ is an isomorphism.  Prove that if $\mathcal{B} = \{\vec{v}_1, \cdots, \vec{v}_n\}$ is a basis for $V$, then $\mathcal{D} = \{h(\vec{v}_1), h(\vec{v}_2), \cdots, h(\vec{v}_n)\}$ is a basis for $W$.

Question 6.  Consider the 4-dimensional vector space $\mathcal{M}_{2\times 2} = \{ \text{ all } 2\times 2 \text{ matrices } \}$ and the function $h:\mathcal{M}_{2\times 2}\to\mathcal{M}_{2\times 2}$ given by $h(M) = M^T$, i.e.

$h\left( \, \left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \,\right) = \left(\begin{array}{cc} a & b \\ c & d \end{array}\right)^T = \left(\begin{array}{cc} a & c \\ b & d \end{array}\right)$.

(a) Prove that $h$ is linear.

(b) Is $h$ one-to-one?  Prove your answer.

(c ) Compute the rank of $h$.

(d) Compute the matrix representation of $h$ relative to the standard basis for $\mathcal{M}_{2\times 2}$ (for both the domain and co-domain).

Question 7.  Let $V$ and $W$ be finite-dimensional vector spaces with bases $\mathcal{B} = \{\vec{v}_1, \cdots, \vec{v}_n\}$ and $\mathcal{D} = \{\vec{w}_1, \cdots \vec{w}_m\}$, respectively.  Prove that the function $\text{Rep}_{\mathcal{B}, \mathcal{D}} : \mathcal{L}(V, W) \to \mathcal{M}_{m\times n}$ given by

$h \mapsto \text{Rep}_{\mathcal{B}, \mathcal{D}}\,(h)$

is an isomorphism of vector spaces.

Question 8.  Let $T:\mathbb{R}^3\to\mathbb{R}^3$ be the function $T(\vec{v}) = A\vec{v}$ where $A$ is the matrix

$\left(\begin{array}{ccc} 1 & 1 & 1 \\ 2 & 0 & 1 \\ 4 & 2 & 3 \end{array}\right)$.

(a) Find a basis for the range space, $\mathcal{R}(T)$.

(b) Find a basis for the null space, $\mathcal{N}(T)$.

(c ) Compute the determinant of $T$.

(d) Are the columns of $T$ linearly independent vectors?  Justify your answer.

(e) Are the rows of $T$ linearly independent vectors?  Justify your answer.

Question 9.  Let $A$ be an $n \times n$ matrix.

(a) Prove that $A$ is non-singular $\iff \mathcal{N}(A) = \{\vec{0}\}$.

(b) Prove that $A$ is non-singular $\iff \mathcal{R}(A) = \mathbb{R}^n$

Question 10.  Consider the function $h:\mathcal{M}_{2\times 3} \to \mathcal{P}_5$ given by

$h\left( \, \left(\begin{array}{ccc} a & b & c \\ d & e & f \end{array}\right)\, \right) = ax^5 + (b+c)x^4 + (a+d)x^3 + ex^2 + (b+c)x + f$.

(a) Just to make sure you understand the function $h$, compute the value of

$h\left( \, \left(\begin{array}{ccc} 1 & 2 & 0 \\ 1 & 1 & 5 \end{array}\right)\, \right)$.

(b) Prove that $h \in \mathcal{L}(\mathcal{M}_{2\times3}, \mathcal{P}_5)$.

(c ) Find a basis for the null space of $h$.

(d) Use only your work in part (c ) to compute the rank of $h$.

(e) Is the polynomial $1 + x + x^2 + x^3 + 2x^4 + x^5$ in the range space $\mathcal{R}(h)$?  Explain your answer.

Question 11.  Let $A$ be a square $n \times n$ matrix.

(a) What does it mean to say that a vector $\vec{v} \in \mathbb{R}^n$ is an eigenvector (with eigenvalue $\lambda$) for $A$?

(b) Let $\lambda$ be an eigenvalue for the matrix $A$, and define the set

$E_{\lambda} = \{ \vec{v} \in \mathbb{R}^n : \vec{v} \text{ is an eigenvector with eigenvalue } \lambda \text{ for } A \} \cup \{\vec{0}\}$.

Prove that $E_{\lambda}$ is a subspace of $\mathbb{R}^n$.

(c ) Prove that $\mathcal{N}(A) = E_{0}$.

Question 12.  (a) Suppose $h: \mathbb{R}^3 \to \mathbb{R}^2$ is a homomorphism and that

$\text{Rep}_{\mathcal{E}_3, \mathcal{E}_2} (h) = \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 7 \end{array}\right)$.

Find a formula for $h(x, y, z)$.

(b) Instead, suppose $f : \mathbb{R}^3\to\mathbb{R}^2$ is a homomorphism and that

$\text{Rep}_{\mathcal{E}_3, \mathcal{D}} (f) = \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 7 \end{array}\right)$

where $\mathcal{D}$ is the basis

$\mathcal{D} = \left\{ \left(\begin{array}{c} 1 \\ 1 \end{array}\right), \left(\begin{array}{c} 3 \\ 4 \end{array}\right) \right\}$

Find a formula for $f(x,y,z)$.

Question 13.  (a) Is the set of vectors

$S = \left\{ \left( \begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right), \left(\begin{array}{c} 1 \\ 1 \\ 0 \end{array}\right), \left(\begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right), \right\}$

a basis for $\mathbb{R}^3$?  Prove your answer.

(b) Is the set of vectors

$S = \left\{ 1 + x + x^2, 1 + x, 1+x^2 \right\}$

a basis for $\mathcal{P}_2$?  Prove your answer.

(c ) Is the $3\times 3$ matrix

$A = \left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right)$

(d) Is the $3 \times 3$ matrix

$A = \left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right)$

(e) Is the set of vectors

$S = \left\{ A\left(\begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right), A\left(\begin{array}{c} 1 \\ 1 \\ 0 \end{array}\right), A\left(\begin{array}{c} 1 \\ 0 \\ 1\end{array}\right) \right\}$

a basis for $\mathbb{R}^3$?  (Here $A$ is the same matrix as in parts (c ) and (d).)  Prove your answer.

(f) Is the determinant of $A$ zero?  Prove / explain your answer.

Question 14.  Let $A$ be the $2\times 2$ matrix

$A = \left( \begin{array}{cc} 1 & 1 \\ 2 & 4 \end{array} \right)$

and consider the set of all $2\times 2$ matrices that “commute” with $A$; that is, consider the set of matrices

$S_A = \left \{ B \in \mathcal{M}_{2\times 2} : AB = BA \right\}$.

(a) Prove that $S_A$ is  a subspace of $\mathcal{M}_{2\times 2}$.

(b) Compute the dimension of $S$.

(c ) Find a $2\times 2$ matrix $M$ so that $S_M = \mathcal{M}_{2\times 2}$ or explain why no such matrix exists.

Question 15.  Assume that $h \in \mathcal{L}(V,W).$

(a) Prove that the range space is a subspace of the codomain $W$.

(b) Prove that the null space is a subspace of $V$.

(c ) Prove that if $U$ is a subspace of the domain $V$, then the set $\{h(\vec{u}) : \vec{u} \in U\}$ (called “the image of $U$ under $h$“) is a subspace of the codomain $W$.  This generalizes the notion of range space.

(d) Suppose $S$ is a subspace of the codomain $W$.  Is it true that the set $\{\vec{v} \in V : h(\vec{v}) \in S \}$ is a subspace of $W$?  Prove your answer.