(a) How does Mathabane's father react to the day's events? (b)How does the conflict between Mathabane's parents reflect their different outlooks on his life?

Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}\right\} \text { and } \mathcal{C}=\left\{\mathbf{c}_{1}, \mathbf{c}_{2}\right\}$ be bases for $\mathbb{R}^{2}$. Find the change-of-coordinates matrix from $\mathcal{B} \text { to } \mathcal{C}$ and the change-of-coordinates matrix from $\mathcal{C} \text { to } \mathcal{B}$.

$$\mathbf{b}_{1}=\left[\begin{array}{l} 4 \\ 4 \end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{l} 8 \\ 4 \end{array}\right], \mathbf{c}_{1}=\left[\begin{array}{l} 2 \\ 2 \end{array}\right], \mathbf{c}_{2}=\left[\begin{array}{r} -2 \\ 2 \end{array}\right]$$

Suppose $f :\mathbb{R}^3 \to \mathbb{R}$ is $\mathcal{C}^2$, and that $\mathbf{x}_0$ is a critical point for $f$. Suppose $Hf (\mathbf{x}_0)(h) = h^2_1+ h_2^2+ h^2_3+ 4h_2h_3$. Does $f$ have a local maximum, minimum, or saddle at $\mathbf{x}_0$?

Determine which of the following linear operators are symmetric. Explain why each is, or is not, symmetric. *(a) $L : \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}$ given by $L=L_{1}^{-1} \circ L_{2} \circ L_{1}$ where $L_{1} : \mathbb{R}^{4} \rightarrow \mathcal{P}_{3}$ is given by $L_{1}([a, b, c, d])=a x^{3}+b x^{2}+c x+d$, and $L_{2} : \mathcal{P}_{3} \rightarrow \mathcal{P}_{3}$ is given by $L_{2}(\mathbf{p}(\mathbf{x}))=\mathbf{p}^{\prime}(x)$. (b) $L : \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by the orthogonal reflection through the plane $5 x+$ $4 y-2 z=0$. (c) $L : \mathbb{R}^{9} \rightarrow \mathbb{R}^{9}$ given by $L=L_{1}^{-1} \circ L_{2} \circ L_{1}$ where $L_{1} : \mathbb{R}^{9} \rightarrow \mathcal{M}_{33}$ is given by

$$L_{1}([a, b, c, d, e, f, g, b, i])=\left[ \begin{array}{lll}{a} & {b} & {c} \\ {d} & {e} & {f} \\ {g} & {b} & {i}\end{array}\right]$$

, and $L_{2} : \mathcal{M}_{33} \rightarrow \mathcal{M}_{33}$ is given by $L_{2}(\mathbf{A})=\mathbf{A}^{T}$.

Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}\right\} \text { and } \mathcal{C}=\left\{\mathbf{c}_{1}, \mathbf{c}_{2}\right\}$ be bases for $\mathbb{R}^{2}$. Find the change-of-coordinates matrix from $\mathcal{B} \text { to } \mathcal{C}$ and the change-of-coordinates matrix from $\mathcal{C} \text { to } \mathcal{B}$.

$$\mathbf{b}_{1}=\left[\begin{array}{r} 6 \\ -12 \end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{l} 4 \\ 2 \end{array}\right], \mathbf{c}_{1}=\left[\begin{array}{l} 4 \\ 2 \end{array}\right], \mathbf{c}_{2}=\left[\begin{array}{l} 3 \\ 9 \end{array}\right]$$

Let $L : \mathcal{V} \rightarrow \mathcal{W}$ be a linear transformation between vector spaces, and let B be a basis for $\mathcal{V}$. (a) Show that if L is an isomorphism, then L(B) is a basis for $\mathcal{W}$. (b) Prove that if L(B) is a basis for $\mathcal{W}$, and the images of vectors in B are distinct, then L is an isomorphism. (c) Define $T : \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ by

$$T(\mathbf{X})=\left[ \begin{array}{lll}{3} & {5} & {3} \\ {1} & {2} & {1}\end{array}\right] \mathbf{X}$$

, and let B be the standard basis in $\mathbb{R}^{3}$. Show that T(B) is a basis for $\mathbb{R}^{2}$, but T is not an isomorphism. (d) Explain why part (c) does not provide a counterexample to part (b).

Let $\mathcal{V}$ and $\mathcal{W}$ be vector spaces. (a) Show that the identity mapping $i : \mathcal{V} \rightarrow \mathcal{V}$ given by $i(\mathbf{v})=\mathbf{v}$, for all $\mathbf{v} \in \mathcal{V}$, is a linear operator. (b) Show that the zero mapping $z : \mathcal{V} \rightarrow \mathcal{W}$ given by $z(\mathbf{v})=\mathbf{0}_{\mathcal{W}}$, for all $\mathbf{v} \in \mathcal{V}$, is a linear transformation.

Let

$$A=\left[ \begin{array}{rr}{1} & {1} \\ {-1} & {3}\end{array}\right]$$

and $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}\right\}$, for

$$\mathbf{b}_{1}=\left[ \begin{array}{l}{1} \\ {1}\end{array}\right]$$

,

$$\mathbf{b}_{2}=\left[ \begin{array}{l}{5} \\ {4}\end{array}\right]$$

. Define $T : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ by T(x)=Ax. a. Verify that $\mathbb{b}_1$ is an eigenvector of A but A is not diagonalizable. b. Find the $\mathcal{B}$-matrix for T.

Define $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ by $T(\mathbf{x})=A \mathbf{x}$. Find a basis $\mathcal{B}$ for $\mathbb{R}^{2}$ with the property that $[T]_{\mathcal{B}}$ is diagonal.

$$A=\left[\begin{array}{rr} 1 & 2 \\ 3 & -4 \end{array}\right]$$

For each of the following linear transformations $L : \mathcal{V} \rightarrow \mathcal{W}$, find a basis for $\operatorname{ker}(L)$ and a basis for $\operatorname{range}(L)$, and verify that $\operatorname{dim}(\operatorname{ker}(L))+\operatorname{dim}(\operatorname{range}(L))=$ $\operatorname{dim}(D)$: *(a) $L : \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ given by $L\left(\left[x_{1}, x_{2}, x_{3}\right]\right)=\left[0, x_{2}\right]$, (b) $L : \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ given by $L\left(\left[x_{1}, x_{2}\right]\right)=\left[x_{1}, x_{1}+x_{2}, x_{2}\right]$, (c) $L : \mathcal{M}_{22} \rightarrow \mathcal{M}_{32}$ given by

$$L\left(\left[ \begin{array}{ll}{a_{11}} & {a_{12}} \\ {a_{21}} & {a_{22}}\end{array}\right]\right)$$

$$=\left[ \begin{array}{cc}{0} & {-a_{12}} \\ {-a_{21}} & {0} \\ {0} & {0}\end{array}\right]$$

, *(d) $L : \mathcal{P}_{4} \rightarrow \mathcal{P}_{2}$ given by $L\left(a x^{4}+b x^{3}+c x^{2}+d x+e\right)=c x^{2}+d x+e$, (e) $L : \mathcal{P}_{2} \rightarrow \mathcal{P}_{3}$ given by $L\left(a x^{2}+b x+c\right)=c x^{3}+b x^{2}+a x$, *(f) $L : \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by $L\left(\left[x_{1}, x_{2}, x_{3}\right]\right)=\left[x_{1}, 0, x_{1}-x_{2}+x_{3}\right]$, *(g) $L : \mathcal{M}_{22} \rightarrow \mathcal{M}_{22}$ given by $L(\mathbf{A})=\mathbf{A}^{T}$, (h) $L : \mathcal{M}_{33} \rightarrow \mathcal{M}_{33}$ given by $L(\mathbf{A})=\mathbf{A}-\mathbf{A}^{T}$, *(i) $L : \mathcal{P}_{2} \rightarrow \mathbb{R}^{2}$ given by $L(\mathbf{p})=\left[\mathbf{p}(1), \mathbf{p}^{\prime}(1)\right]$, (j) $L : \mathcal{P}_{4} \rightarrow \mathbb{R}^{3}$ given by $L(\mathbf{p})=[\mathbf{p}(-1), \mathbf{p}(0), \mathbf{p}(1)]$.

Define $f(x)=x^{2}$ for x in $\mathbb{R}$. a. Show that if $\mathcal{O}$ is any open subset of $\mathbb{R}$ that does not contain 0, then $f(\mathcal{O})$ is open. b. Show that if $\mathcal{O}$ is an open subset of $\mathbb{R}$ that contains 0, then $f(\mathcal{O})$ is not open.

Suppose that $\Psi: \mathcal{O} \rightarrow \mathbb{R}^{n}$ is a smooth change of variables on the open subset $\mathcal{O}$ of $\mathbb{R}^{n}$ and that K is a closed bounded subset of $\mathcal{O}$. Prove that there is a positive number c such that for any two points $\mathbf{u}$ and $\mathbf{v}$ in K

$$\|\Psi(\mathbf{u})-\Psi(\mathbf{v})\| \geq c\|\mathbf{u}-\mathbf{v}\|$$

Let $\mathcal{V}$ be any vector space, and let $\mathcal{W}$ be a nonempty subset of $\mathcal{V}$.

(a) Prove that $\mathcal{W}$ is a subspace of $\mathcal{V}$ if and only if $a \mathbf{w}_1+b \mathbf{w}_2$ is an element of $\mathcal{W}$ for every $a, b \in \mathbb{R}$ and every $\mathbf{w}_1, \mathbf{w}_2 \in \mathcal{W}$. (Hint: For one half of the proof, first consider the case where $a=b=1$ and then the case where $b=0$ and $a$ is arbitrary.)

(b) Prove that $\mathcal{W}$ is a subspace of $\mathcal{V}$ if and only if $a \mathbf{w}_1+\mathbf{w}_2$ is an element of $\mathcal{W}$ for every real number $a$ and every $\mathbf{w}_1$ and $\mathbf{w}_2$ in $\mathcal{W}$.

Let $\mathcal{E}=\left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right\}$ be the standard basis for $\mathbb{R}^{3}$, let $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}\right\}$ be a basis for a vector space V, and let $T: \mathbb{R}^{3} \rightarrow V$ be a linear transformation with the property that $T\left(x_{1}, x_{2}, x_{3}\right)=\left(2 x_{3}-x_{2}\right) \mathbf{b}_{1}-\left(2 x_{2}\right) \mathbf{b}_{2}+\left(x_{1}+3 x_{3}\right) \mathbf{b}_{3}$ a. Compute $T\left(\mathbf{e}_{1}\right), T\left(\mathbf{e}_{2}\right), \text { and } T\left(\mathbf{e}_{3}\right)$. b. Compute $\left[T\left(\mathbf{e}_{1}\right)\right]_{\mathcal{B}},\left[T\left(\mathbf{e}_{2}\right)\right]_{\mathcal{B}}, \text { and }\left[T\left(\mathbf{e}_{3}\right)\right]_{\mathcal{B}}$. c. Find the matrix for T relative to $\mathcal{E} \text { and } \mathcal{B}$.