Problems

01

Linearity

Suppose that ${\mathbf{u}}_1,\dots ,{\mathbf{u}}_k\in {\mathbb{R}}^n$ are vectors and that $A$ represents some linear transformation acting on ${\mathbb{R}}^n$.

• (a) Suppose we know that ${\mathbf{u}}_1,\dots ,{\mathbf{u}}_k$ are independent. Do we automatically know that $A{\mathbf{u}}_1,\dots ,A{\mathbf{u}}_k$ are independent? (Justify your answer. If false, a counterexample suffices; if true, an argument is required.)
• (b) Suppose we know that ${\mathbf{u}}_1,\dots ,{\mathbf{u}}_k$ are dependent. Do we automatically know that $A{\mathbf{u}}_1,\dots ,A{\mathbf{u}}_k$ are dependent? (Justify your answer. If false, a counterexample suffices; if true, an argument is required.)
• (c) If $\vec{\ell }(t)$ parametrizes a line passing through the origin, do we automatically know that $A\vec{\ell }$ is also a line passing through the origin? What if we know that $A$ is invertible?
• (d) If $\vec{\ell }(t)$ parametrizes a line not passing through the origin, do we automatically know that $A\vec{\ell }$ is also a line not passing through the origin? What if we know that $A$ is invertible?

02

Changing bases

• (a) Suppose that $\mathcal{B}=\{{\mathbf{b}}_1,{\mathbf{b}}_2,{\mathbf{b}}_3\}$ and $\mathcal{C}=\{{\mathbf{c}}_1,{\mathbf{c}}_2,{\mathbf{c}}_3\}$ are bases of $V$. Suppose that $A=\left(\begin{array}{rrr}[{\mathbf{c}}_1{]}_{\mathcal{B}}& [{\mathbf{c}}_2{]}_{\mathcal{B}}& [{\mathbf{c}}_3{]}_{\mathcal{B}}\end{array}\right)$ is the matrix with column vectors equal to ${\mathbf{c}}_1,{\mathbf{c}}_2,{\mathbf{c}}_3$ when written in the basis $\mathcal{B}$. Which of the following is satisfied by $A$? (i) $A[\mathbf{x}{]}_{\mathcal{B}}=[\mathbf{x}{]}_{\mathcal{C}}$ or (ii) $A[\mathbf{x}{]}_{\mathcal{C}}=[\mathbf{x}{]}_{\mathcal{B}}$. (You must justify your answer.)
• (b) Find the change of coordinates transfer matrices ${}_{\mathcal{C}}{T}_{\mathcal{B}}$ and ${}_{\mathcal{B}}{T}_{\mathcal{C}}$ between the following two bases of ${\mathbb{R}}^2$: $$\mathcal{B}=\left\{{\mathbf{b}}_1=\left(\begin{array}{c}-1\\ 8\end{array}\right),{\mathbf{b}}_2=\left(\begin{array}{c}1\\ -5\end{array}\right)\right\},\mathcal{C}=\left\{{\mathbf{c}}_1=\left(\begin{array}{c}1\\ 4\end{array}\right),{\mathbf{c}}_2=\left(\begin{array}{c}1\\ 1\end{array}\right)\right\}.$$
• (c) Assume that $\mathcal{B}=\{{\mathbf{b}}_1,{\mathbf{b}}_2,{\mathbf{b}}_3\}$ and $\mathcal{C}=\{{\mathbf{c}}_1,{\mathbf{c}}_2,{\mathbf{c}}_3\}$ are bases of $V$, and that $$\begin{array}{rl}{\mathbf{b}}_1& =2{\mathbf{c}}_1-{\mathbf{c}}_2+{\mathbf{c}}_3\\ {\mathbf{b}}_2& =3{\mathbf{c}}_2+{\mathbf{c}}_3\\ {\mathbf{b}}_3& =-3{\mathbf{c}}_1+2{\mathbf{c}}_3.\end{array}$$ Find the change of basis transfer matrix ${}_{\mathcal{C}}{T}_{\mathcal{B}}$, and also compute $[\mathbf{x}{]}_{\mathcal{C}}$ for the vector $\mathbf{x}={\mathbf{b}}_1-2{\mathbf{b}}_2+2{\mathbf{b}}_3$.

03

Rank-Rank and Rank-Nullity counting practice

Answer the following questions using the Rank-Rank and Rank-Nullity theorems. It may help to work with imaginary matrices in RREF and think about the pivots.

• (a) Suppose $A$ is $3\times 8$ with $\mathrm{rank}A=2$. What is $\dim \mathrm{Ker}(A)$ and $\dim \mathrm{CoKer}(A)$?
• (b) Suppose $A$ is $7\times 3$ with $\mathrm{rank}A=3$. What is $\dim \mathrm{Ker}(A)$ and $\dim \mathrm{CoKer}(A)$?
• (c) Suppose $A$ is $9\times 5$ with $\dim \mathrm{Ker}(A)=2$. What is $\mathrm{rank}A$ and $\dim \mathrm{CoKer}(A)$?

04

Bases and dimensions from row reduction

Row reduce the following matrix $A$ in order to find a basis for the row space $\mathrm{Im}(A^{\mathsf{T}})$, and another basis for the column space $\mathrm{Im}(A)$, and a third basis for the kernel $\mathrm{Ker}(A)$. By counting basis elements to determine dimensions, verify the two theorems of the section about rank.

$$A=\left(\begin{array}{rrrrr}-2& -5& 8& 0& -17\\ 1& 3& -5& 1& 5\\ 3& 11& -19& 7& 1\\ 1& 7& -13& 5& -3\end{array}\right)$$

(Hint: for $\mathrm{Im}(A)$, you should determine which columns of $QA$ have pivots; the same columns of $A$ will then be independent vectors! This happens because row reduction $Q$ preserves independence / dependence relations among the column vectors of $A$.)