Determine:

$$\int _ { \mathcal { C } _ { 2 } } \mathbf { F } \cdot d \mathbf { r }$$

This exercise concerns the set of real continuous functions on $[a, b]$. (a) Let a and b be fixed real numbers with $a<b$, and let $\mathcal{V}$ be the set of all real continuous functions on $[a, b]$. Define $\langle,\rangle$ on $\mathcal{V}$ by $\langle\mathbf{f}, \mathbf{g}\rangle=\int_{a}^{b} \mathbf{f}(t) \mathbf{g}(t) d t$. Prove that this operation is a real inner product on $\mathcal{V}$. *(b) For the inner product of part (a) with $a=0$ and $b=\pi$, find $\langle\mathbf{f}, \mathbf{g}\rangle$ and $\|\mathbf{f}\|$, for $\mathbf{f}=e^{t}$ and $\mathbf{g}=\sin t$.

Use mathematical induction to prove the following statements:

$n^3-n$ is a multiple of 6 or all $n \in \mathbb{Z}^{+}$.

Which of the following linear transformations are one-to-one? Which are onto? Justify your answers without using row reduction. *(a) $L : \mathbb{R}^{3} \rightarrow \mathbb{R}^{4}$ given by $L([x, y, z])=[y, z,-y, 0]$, (b) $L : \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ given by $L([x, y, z])=[x+y, y+z]$, *(c) $L : \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by $L([x, y, z])=[2 x, x+y+z,-y]$, (d) $L : \mathcal{P}_{3} \rightarrow \mathcal{P}_{2}$ given by $L\left(a x^{3}+b x^{2}+c x+d\right)=a x^{2}+b x+c$, *(e) $L : \mathcal{P}_{2} \rightarrow \mathcal{P}_{2}$ given by $L\left(a x^{2}+b x+c\right)=(a+b) x^{2}+(b+c) x+(a+c)$, (f) $L : \mathcal{M}_{22} \rightarrow \mathcal{M}_{22}$ given by

$$L\left(\left[ \begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]\right)=\left[ \begin{array}{cc}{d} & {b+c} \\ {b-c} & {a}\end{array}\right]$$

, *(g) $L : \mathcal{M}_{23} \rightarrow \mathcal{M}_{22}$ given by

$$L\left(\left[ \begin{array}{lll}{a} & {b} & {c} \\ {d} & {e} & {f}\end{array}\right]\right)=\left[ \begin{array}{ll}{a} & {-c} \\ {2 e} & {d+f}\end{array}\right]$$

, *(h) $L : \mathcal{P}_{2} \rightarrow \mathcal{M}_{22}$ given by

$$L\left(a x^{2}+b x+c\right)=\left[ \begin{array}{cc}{a+c} & {0} \\ {b-c} & {-3 a}\end{array}\right]$$

.

True or False: (a) If $L : \mathcal{V} \rightarrow \mathcal{V}$ is a linear operator and $\lambda$ is an eigenvalue for L, then $E_{\lambda}=$ $\{\lambda L(\mathbf{v}) | \mathbf{v} \in \mathcal{V}\}$. (b) If L is a linear operator on a finite dimensional vector space $\mathcal{V}$ and A is a matrix for L with respect to some ordered basis for $\mathcal{V}$, then $p_{L}(x)=p_{\mathbf{A}}(x)$. (c) If $\operatorname{dim}(\mathcal{V})=5$, a linear operator L on $\mathcal{V}$ is diagonalizable when L has 5 linearly independent eigenvectors. (d) Eigenvectors for a given linear operator L are linearly independent if and only if they correspond to distinct eigenvalues of L. (e) If L is a linear operator on a finite dimensional vector space, then the union of bases for distinct eigenspaces for L is a linearly independent set. (f) If $L : \mathbb{R}^{6} \rightarrow \mathbb{R}^{6}$ is a diagonalizable linear operator, then the union of bases for all the distinct eigenspaces of L is actually a basis for $\mathbb{R}^{6}$. (g) If L is a diagonalizable linear operator on a finite dimensional vector space $\mathcal{V}$, the Generalized Diagonalization Method produces a basis B for $\mathcal{V}$ so that the matrix for L with respect to B is diagonal. (h) If L is a linear operator on a finite dimensional vector space $\mathcal{V}$ and $\lambda$ is an eigenvalue for L, then the algebraic multiplicity of $\lambda$ is never greater than the geometric multiplicity of $\lambda$. (i) If $\operatorname{dim}(\mathcal{V})=7$ and $L : \mathcal{V} \rightarrow \mathcal{V}$ is a linear operator, then L is diagonalizable whenever the sum of the algebraic multiplicities of all the eigenvalues equals 7. (j) If

$$\mathbf{A}=\left[ \begin{array}{ll}{1} & {2} \\ {0} & {4}\end{array}\right]$$

, then $\left(1 \mathbf{I}_{2}-\mathbf{A}\right)\left(4 \mathbf{I}_{2}-\mathbf{A}\right)=\mathbf{O}_{2}$.

Let $S=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ be a linearly independent subset of $\mathbb{R}^{3}$. Find a linear operator $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$, such that $\left\{T\left(\mathbf{v}_{1}\right), T\left(\mathbf{v}_{2}\right), T\left(\mathbf{v}_{3}\right)\right\}$ is linearly dependent.

In each case, show that T is not a linear transformation. (a)

$$T : \mathbf { M } _ { n n } \rightarrow \mathbb { R } ; T ( A ) = \operatorname { det } A$$

, (b)

$$T : \mathbf { M } _ { n m } \rightarrow \mathbb { R } ; T ( A ) = \operatorname { rank } A$$

, (c)

$$T : \mathbb { R } \rightarrow \mathbb { R } ; T ( x ) = x ^ { 2 }$$

, (d)

$$T : V \rightarrow V ; T ( \mathbf { v } ) = \mathbf { v } + \mathbf { u }$$

where

$$\mathbf { u } \neq \mathbf { 0 }$$

is a fixed vector in V (T is called the translation by u)

For a metric space X and a positive number r, can one have $\mathcal{B}_{r}(p)=\mathcal{B}_{r}(q)$ and yet $p \neq q$? Can this happen in $\mathbb{R}^{n}$ with the Euclidean metric?

(a) Find the coordinate vectors

$$[ \mathbf { x } ] _ { \mathcal { B } } \text { and } [ \mathbf { x } ] _ { \mathcal { C } }$$

of

$$\mathbf x$$

with respect to the bases

$$\mathcal { B } \text { and } \mathcal { C },$$

respectively. (b) Find the change-of-basis matrix

$$P _ { \mathcal { C } \leftarrow \mathcal { B } } \text { from } \mathcal { B } \text { to } \mathcal { C }.$$

(c) Use your answer to part (b) to compute

$$[ \mathbf { x } ] _ { \mathcal { C } },$$

and compare your answer with the one found in part (a). (d) Find the change-of-basis matrix

$$P _ { B \leftarrow C } \text { from } \mathcal { C } \text { to } \mathcal { B }.$$

(e) Use your answers to parts (c) and (d) to compute

$$[ \mathbf { x } ] _ { \mathcal { B } },$$

and compare your answer with the one found in part (a).

$$\mathbf { x } = \left[ \begin{array} { l } { 3 } \\ { 1 } \\ { 5 } \end{array} \right] , \mathcal { B } = \left\{ \left[ \begin{array} { l } { 0 } \\ { 1 } \\ { 0 } \end{array} \right] , \left[ \begin{array} { l } { 0 } \\ { 0 } \\ { 1 } \end{array} \right] , \left[ \begin{array} { l } { 1 } \\ { 0 } \\ { 0 } \end{array} \right] \right\}, \mathcal { C } = \left\{ \left[ \begin{array} { l } { 1 } \\ { 1 } \\ { 0 } \end{array} \right] , \left[ \begin{array} { l } { 0 } \\ { 1 } \\ { 1 } \end{array} \right] , \left[ \begin{array} { l } { 1 } \\ { 0 } \\ { 1 } \end{array} \right] \right\} \text { in } \mathbb { R } ^ { 3 }$$

Define $T \in \mathcal{L}\left(\mathcal{P}_{4}(\mathbf{R})\right)$ by $(T p)(x)=x p^{\prime}(x)$ for all $x \in \mathbf{R}.$ Find all eigenvalues and eigenvectors of T.

Find a polynomial $q \in \mathcal{P}_{2}(\mathbf{R})$ such that $p\left(\frac{1}{2}\right)=\int_{0}^{1} p(x) q(x) d x$ for every $p \in \mathcal{P}_{2}(\mathbf{R}).$

Let $\mathcal{O}$ be an open subset of $\mathbb{R}^{n}$ and suppose that the function $f: \mathcal{O} \rightarrow \mathbb{R}$ is continuous. If a and b are numbers with a < b, prove that the set

$$\{\mathbf{u} \text { in } \mathcal{O} | a<f(\mathbf{u})<b\}$$

is open in $\mathbb{R}^{n}$.

Under what conditions are the following true for vectors

$$\mathbf { u } \text { and } \mathbf { v } \text { in } \mathbb { R } ^ { 2 } \text { or } \mathbb { R } ^ { 3 } \text { ? }$$

$$\text { (a) } \| \mathbf { u } + \mathbf { v } \| = \| \mathbf { u } \| + \| \mathbf { v } \|\quad \text { (b) } \| \mathbf { u } + \mathbf { v } \| = \| \mathbf { u } \| - \| \mathbf { v } \|$$

Let $\mathcal{V}$ be an n-dimensional complex vector space with basis $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{n}\right\}$. Prove that $\left\{\mathbf{v}_{1}, i \mathbf{v}_{1}, \mathbf{v}_{2}, i \mathbf{v}_{2}, \dots, \mathbf{v}_{n}, i \mathbf{v}_{n}\right\}$ is a basis for $\mathcal{V}$ when considered as a real vector space.