Section 1.1: 4, 12, 16, 18, 20
Section 1.2: 4, 14, 16

Section 1.4: 10, 12, 14
Section 2.1: 2, 6, 10, 12, 20

Section 1.5: 8, 20, 22
Section 1.7: 2, 10, 32

Extra:
1. Let $A$ be the $3\times 3$ zero matrix. Express the solution to $A{\bf x}={\bf 0}$ in parametric vector form.
2. Explain why, if $A$ is an $m\times n$ matrix and $m\lt n$, the columns of $A$ must be dependent.

Section 1.8: 18, 30
Section 1.9: 4, 6, 8, 10, 12, 30

Extra: 1. If ${\bf p}$ and ${\bf d}$ are vectors in $\mathbb R^n$, and ${\bf d}\neq {\bf 0}$, then the set of all vectors of the form ${\bf x}={\bf p}+t{\bf d}$, $t\in \mathbb R$, is a line in ${\mathbb R}^n$. If ${\bf d}={\bf 0}$, then this set is just the singleton $\{{\bf p}\}$.
Show that the image of this line under a linear transformation is either a line or a singleton.

Section 1.6: 6
Section 2.1: 26, 28
Section 2.2: 2, 16, 20, 22, 24

Extra:
1. Show that if $A$ has a left inverse, then $A^t$ has a right inverse.

Section 2.3: 8, 18
Section 2.8: 2, 22(a)(b)(d), 24
Section 2.9: 14, 18(a)(c)(d)

Extra:
1. Explain why $\mathbb R^4$ has no $5$-dimensional subspace.

Section 3.1: 4, 14, 16, 40
Section 3.2: 8, 22, 24, 34
Section 3.3: 22

Section 4.1: 22, 24, 28
Section 4.2: 26
Section 4.3: 22
Section 4.4: 14, 16, 32
Extra:
1. Find a basis for the kernel of the linear transformation $L\colon \mathbb P_2\to \mathbb R^2\colon p(t)\mapsto [p(0)\;p'(0)]^T$.
(The kernel of a linear transformation $L$ is the set of all $\mathbf x$ such that $L({\mathbf x}) = {\mathbf 0}$ Observe that any correct answer to this problem will be a set of polynomials.)

Section 4.5: 32
Section 4.6: 4, 8, 30
Section 4.7: 6, 8
Chapter 4 Supplementary Exercises: 4, 10, 12
Extra:
1. Suppose that $\mathbb V\leq \mathbb R^3$ is spanned by ${\mathcal B} = \left( \left[\begin{array}{c}1\\2\\3\end{array}\right], \left[\begin{array}{c}4\\5\\6\end{array}\right] \right)$ and $\mathbb W\leq \mathbb R^3$ is spanned by ${\mathcal C} = \left(\left[\begin{array}{c}1\\2\\4\end{array}\right], \left[\begin{array}{c}1\\3\\9\end{array}\right] \right)$. Find bases for $\mathbb V+\mathbb W$ and $\mathbb V\cap\mathbb W$.

Section 5.1: 8, 16, 22(a)(b)(d)(e), 30
Section 5.2: 20
Section 5.3: 8, 12

Extra:
1. Show that if $\lambda$ is an e-value for $A$ and $k$ is an integer, then $\lambda^k$ is an e-value for $A^k$.

2. Suppose that $A$ is a block upper triangular matrix with diagonal blocks $A_1,\ldots,A_k$. Show that the characteristic polynomial of $A$ is the product of the characteristic polynomials of the $A_i$'s.

Section 5.4: 14, 19, 20, 22
Section 5.5: 2
Extra:
1. Suppose that $A$ is a $2\times 2$-matrix and that $\det(A-I) = -16$ and $\det(A-2I) = -15$. What are the e-values of $A$?

2. Let $\mathbb P_n(t)$ be the space of polynomials of degree at most $n$ in the variable $t$. Let $D\colon\mathbb P_3(t)\to \mathbb P_3(t)\colon f\mapsto f'$ be the operation of differentiation. Find the characteristic polynomial, e-values, and e-spaces of $D$.

3. Let $T\colon M_{2\times 2}(\mathbb R)\to M_{2\times 2}(\mathbb R)\colon M\mapsto M^T$ be the operation of transpose. Find the characteristic polynomial, e-values, and e-spaces of $T$.

4. Let $E(t)$ be the subspace of $\mathbb P_3(t)$ consisting of polynomials whose monomials have even exponent (like $1$, $t^2$, $t^4$, etc) and let $O(t)$ be the subspace consisting of polynomials whose monomials have odd exponent (like $t$, $t^3$, etc). Show that $\mathbb P_n(t) = E(t) \oplus O(t)$. What are the dimensions of $\mathbb P_n(t)$, $E(t)$ and $O(t)$ when $n=100$?

5. Suppose that $T\colon V\to V$ is a linear transformation defined on a finite dimensional vector space $V$ and that $T$ has an eigenvalue $\lambda$ whose associated eigenspace is all of $V$ (i.e. $V_{\lambda} = V$). What are the possible matrices ${}_{\mathcal B}[T]_{\mathcal B}$ for different bases ${\mathcal B}$?

Section 5.5: 2
Chapter 5, Supplementary Problems: 6, 10
Extra:
1. What is the minimal polynomial of the $3\times 3$ matrix of all $1$'s? Does your answer suggest that the matrix is diagonalizable or nondiagonalizable?

2. How is the minimal polynomial of $A$ related to the minimal polynomial of $A-I$?

3. Show that if $A$ is diagonalizable, then $A-I$ is diagonalizable.