Practice problems (pages 9-12): 1.17, 1.18, 1.23, 1.36.

Problems to turn in:

1. Exercise 1.22.

2. Exercise 1.27.

3. Find all triples $(x,y,z)$ of positive real numbers satisfying the nonlinear system

$$ \begin{array}{rl} x^2y&=2\\ xy^2z&=4\\ yz^2&=8 \end{array} $$

using the following method. First, take the logarithm of each equation. Then replace $\log(x)$ with a new symbol $X$, $\log(y)$ with $Y$, $\log(z)$ with $Z$, and solve the resulting linear system for $(X, Y, Z)$. Use your answer to solve the original system. (You can use a log to any base, but it is probably easiest to use base $2$.)

Solution sketches (linalghw1sol.tex),
Solution sketches (linalghw1sol.pdf),
Solution sketches (video).

Problems to turn in:

1. Let $A = \begin{bmatrix} 1&1\\ 0&1\end{bmatrix}$. Show by induction that $A^n = \begin{bmatrix} 1& n\\ 0&1\end{bmatrix}$.

2. I have three matrices: $A$ is $m\times n$, $B$ is $p\times q$, and $C$ is $r\times s$. Suppose I tell you that, because of their dimensions, the triple product $ABC$ is not defined, even though the triple product in every other order is defined (i.e., $ACB, BAC, BCA, CAB, CBA$ are all defined). Explain how you know that I made a mistake.

3. This problem is about multiplicative inverses of matrices.
(a) Find the multiplicative inverse of each type of elementary matrix: $P_{ij}$, and $E_{ij}(r)$ for the case $i\neq j$ and the case $i=j$.
(b) Show that if $A^{-1}$ is the multiplicative inverse of $A$ and $B^{-1}$ is the multiplicative inverse of $B$, then $B^{-1}A^{-1}$ is the multiplicative inverse of $AB$.

Solution sketches (linalghw2sol.tex),
Solution sketches (linalghw2sol.pdf),

Practice problems (pages 46-47): 2.11, 2.12, 2.17 (Hint: Yes. Why?).

Problems to turn in:

1. Exercise 2.16, parts (b), (c), (d).

2. Let ${\bf e}_i$ be the vector of length $n$ whose $i$th entry is $1$ and whose other entries are $0$. The set $\{{\bf e}_1, {\bf e}_2, \ldots, {\bf e}_n\}$ is called the ``standard basis for $\mathbb R^n$''. Show that the standard basis is independent.

3. Let $A$ be an $m\times n$ matrix and let ${\bf e}_i$ be a standard basis vector of length $n$. Explain why the product $A\cdot {\bf e}_i$ equals the $i$th column of $A$.

Optional Fun Challenge! (0 points!)
An ant starts at a point in the plane and walks in a straight line for 1 unit. He then turns left at a right angle and walks in a straight line for 1/2 unit. He then turns left again at a right angle and walks 1/3 unit. He continues to turn left at right angles and walk 1/4 unit, 1/5 unit, 1/6 unit, ETC. As time progresses, this ant spirals closer and closer to a limiting location. At the limit, how far will the ant be from where he started?

Solution sketches (linalghw3sol.tex),
Solution sketches (linalghw3sol.pdf),

Practice problems (pages 92-96): 1.17, 1.20, 1.22(b).

Problems to turn in:

1. Describe a way to view the set $\mathbb C$ of complex numbers as a real vector space. (This means: describe the additive structure and how to scale elements by a real number.)

2. Let $\mathbb R[x]$ be the real vector space of polynomials in the variable $x$ with real coefficients. Let $S\subseteq \mathbb R[x]$ be the subset of polynomials whose roots are real numbers. Determine whether $S$ is a subspace of $\mathbb R[x]$. (Say the answer, then give the reason.)

3. Let $\mathbb R[x]$ be the real vector space of polynomials in the variable $x$ with real coefficients. Let $Z\subseteq \mathbb R[x]$ be the subset of those polynomials with roots at $x=0, 1, 2$. Determine whether $Z$ is a subspace of $\mathbb R[x]$. (Say the answer, then give the reason.)

Solution sketches (linalghw4sol.tex),
Solution sketches (linalghw4sol.pdf),

This set of exercises will concern the problem of determining the matrix of a linear transformation relative to input and output bases. We are using the notation ${}_{\mathcal C}[T]_{\mathcal B}$, introduced in class on Friday, March 5. The book uses a slightly different notation for this: $[T]_{{\mathcal B},{\mathcal C}}$, see the definition at the top of page 214 and Remark 1.3 below it.

Problems to turn in:

1. Let $P_n$ be the $\mathbb R$-vector space of polynomials of degree at most $n$. Let ${\mathcal B}=(1,x,\ldots,x^n)$ be an ordered basis for this space. Let $D\colon P_n\to P_n\colon f(x)\mapsto f'(x)$ be the differentiation map (which is linear). Determine ${}_{\mathcal B}[D]_{\mathcal B}$.

2. Using the notation of Problem 1, let $T_n\colon P_n\to P_n$ be defined by $T_n(f(x)) = f(x+1)$. ($T_n$ is linear.)
(a) Determine ${}_{\mathcal B}[T_3]_{\mathcal B}$.
(b) Predict what ${}_{\mathcal B}[T_n]_{\mathcal B}$ will look like. (You don't have to verify your guess.)

3. Explain why
(a) ${}_{\mathcal D}[T]_{\mathcal C}\cdot {}_{\mathcal C}[S]_{\mathcal B}= {}_{\mathcal D}[T\circ S]_{\mathcal B}$.
(b) ${}_{\mathcal B}[\textrm{id}]_{\mathcal B}=I$.
(Here ``$\textrm{id}$'' refers to the ``identity transformation'' which is the transformation $\textrm{id}(x)=x$.)

Solution sketches (linalghw5sol.tex),
Solution sketches (linalghw5sol.pdf),

3/19/21

2. Let $A$ be the matrix for differentiation $D\colon P_3\to P_3 \colon f(x)\mapsto f'(x)$ relative to the ordered basis $(1, x, x^2, x^3)$, and let $T(x)=Ax$.
(a) Find a basis for the image of $T$.
(b) Find a basis for the kernel of $T$.
(c) What is the rank of $T$?

3. If $A$ is an $n\times n$ matrix, then any matrix of the form $B = C^{-1}AC$ is called a conjugate of $A$. Explain why the following statement is true: If $T\colon \mathbb R^n\to \mathbb R^n$ is a linear transformation whose matrix relative to the standard basis $\mathcal E$ is $A = {}_{\mathcal E}[T]_{\mathcal E}$, then the matrix ${}_{\mathcal B}[T]_{\mathcal B}$ for $T$ relative to some other basis $\mathcal B$ is a conjugate of the matrix $A$.

Solution sketches (linalghw6sol.tex),
Solution sketches (linalghw6sol.pdf).

4/2/21

2. Let $P_3$ be the $4$-dimensional real vector space of all polynomials in $\mathbb R[x]$ that have degree at most $3$. Let $S\leq P_3$ be the $2$-dimensional subspace of those polynomials $p(x)$ satisfying $p(1)=p(2)=0$. Find bases for $P_3$ and $S$, and a basis for a complement $S^{\perp}$ to $S$.

3. Use a determinant to find the area of the triangle in $\mathbb R^2$ whose vertices are $(1,1), (3,61), (101,3)$. (Hint: a triangle is half a parallelogram.)

Solution sketches (linalghw7sol.tex),
Solution sketches (linalghw7sol.pdf).

4/9/21

2. Explain why the determinant of an upper triangular matrix equals the product of the diagonal entries.

3. Let $A$ and $B$ be $n\times n$ matrices with $B = [{\bf b}_1 \cdots {\bf b}_n]$.
(a) Use the fact that $AB = [A{\bf b}_1 \cdots A{\bf b}_n]$ to show that, as a function of the columns of $B$, $f({\bf b}_1,\ldots,{\bf b}_n):=\det(AB)$ is multilinear and alternating.
(b) We showed that if $f$ is a multilinear and alternating function of $n$ columns, then $$f(B) = \textrm{(perm expansion of $\det(B)$)}\cdot f({\bf e}_1,\ldots,{\bf e}_n).$$ Use this and part (a) to conclude that $\det(AB)=\det(B)\cdot\det(A)=\det(A)\cdot\det(B)$.

Solution sketches (linalghw8sol.tex),
Solution sketches (linalghw8sol.pdf).

2. 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 $A$?

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 $T$.

Solution sketches (linalghw9sol.tex),
Solution sketches (linalghw9sol.pdf).

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.
Solution sketches (linalghw10sol.tex),
Solution sketches (linalghw10sol.pdf).