## Problem 706

Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set
$S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}$ still a spanning set for $V$? If so, prove it. Otherwise, give a counterexample.

## Problem 705

For a set $S$ and a vector space $V$ over a scalar field $\K$, define the set of all functions from $S$ to $V$
$\Fun ( S , V ) = \{ f : S \rightarrow V \} .$

For $f, g \in \Fun(S, V)$, $z \in \K$, addition and scalar multiplication can be defined by
$(f+g)(s) = f(s) + g(s) \, \mbox{ and } (cf)(s) = c (f(s)) \, \mbox{ for all } s \in S .$

(a) Prove that $\Fun(S, V)$ is a vector space over $\K$. What is the zero element?

(b) Let $S_1 = \{ s \}$ be a set consisting of one element. Find an isomorphism between $\Fun(S_1 , V)$ and $V$ itself. Prove that the map you find is actually a linear isomorpism.

(c) Suppose that $B = \{ e_1 , e_2 , \cdots , e_n \}$ is a basis of $V$. Use $B$ to construct a basis of $\Fun(S_1 , V)$.

(d) Let $S = \{ s_1 , s_2 , \cdots , s_m \}$. Construct a linear isomorphism between $\Fun(S, V)$ and the vector space of $n$-tuples of $V$, defined as
$V^m = \{ (v_1 , v_2 , \cdots , v_m ) \mid v_i \in V \mbox{ for all } 1 \leq i \leq m \} .$

(e) Use the basis $B$ of $V$ to constract a basis of $\Fun(S, V)$ for an arbitrary finite set $S$. What is the dimension of $\Fun(S, V)$?

(f) Let $W \subseteq V$ be a subspace. Prove that $\Fun(S, W)$ is a subspace of $\Fun(S, V)$.

## Problem 704

Let $A=\begin{bmatrix} 2 & 4 & 6 & 8 \\ 1 &3 & 0 & 5 \\ 1 & 1 & 6 & 3 \end{bmatrix}$.
(a) Find a basis for the nullspace of $A$.

(b) Find a basis for the row space of $A$.

(c) Find a basis for the range of $A$ that consists of column vectors of $A$.

(d) For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.

## Problem 703

Using the definition of the range of a matrix, describe the range of the matrix
$A=\begin{bmatrix} 2 & 4 & 1 & -5 \\ 1 &2 & 1 & -2 \\ 1 & 2 & 0 & -3 \end{bmatrix}.$

## Problem 702

The following problems are True or False.

Let $A$ and $B$ be $n\times n$ matrices.

(a) If $AB=B$, then $B$ is the identity matrix.
(b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions.
(c) If $A$ is invertible, then $ABA^{-1}=B$.
(d) If $A$ is an idempotent nonsingular matrix, then $A$ must be the identity matrix.
(e) If $x_1=0, x_2=0, x_3=1$ is a solution to a homogeneous system of linear equation, then the system has infinitely many solutions.

## Problem 701

Find the vector form solution $\mathbf{x}$ of the equation $A\mathbf{x}=\mathbf{0}$, where $A=\begin{bmatrix} 1 & 1 & 1 & 1 &2 \\ 1 & 2 & 4 & 0 & 5 \\ 3 & 2 & 0 & 5 & 2 \\ \end{bmatrix}$. Also, find two linearly independent vectors $\mathbf{x}$ satisfying $A\mathbf{x}=\mathbf{0}$.

## Problem 700

Let $A$ be an $n\times n$ nonsingular matrix. Let $\mathbf{v}, \mathbf{w}$ be linearly independent vectors in $\R^n$. Prove that the vectors $A\mathbf{v}$ and $A\mathbf{w}$ are linearly independent.

## Problem 699

(a) Find a $3\times 3$ nonsingular matrix $A$ satisfying $3A=A^2+AB$, where $B=\begin{bmatrix} 2 & 0 & -1 \\ 0 &2 &-1 \\ -1 & 0 & 1 \end{bmatrix}.$

(b) Find the inverse matrix of $A$.

## Problem 698

Let $A$ and $B$ be $3\times 3$ matrices and let $C=A-2B$.
If
$A\begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}=B\begin{bmatrix} 2 \\ 6 \\ 10 \end{bmatrix},$ then is the matrix $C$ nonsingular? If so, prove it. Otherwise, explain why not.

## Problem 697

Find all $2\times 2$ symmetric matrices $A$ satisfying $A\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$? Express your solution using free variable(s).

## Problem 696

Let
$A=\begin{bmatrix} -4 & -6 & -12 \\ -2 &-1 &-4 \\ 2 & 3 & 6 \end{bmatrix}, \quad \mathbf{u}=\begin{bmatrix} 6 \\ 5 \\ -3 \end{bmatrix}, \quad \mathbf{v}=\begin{bmatrix} -2 \\ 0 \\ 1 \end{bmatrix}, \quad \text{ and } \mathbf{w}=\begin{bmatrix} -2 \\ -1 \\ 1 \end{bmatrix}.$

(a) Express the vector $\mathbf{u}$ as a linear combination of $\mathbf{v}$ and $\mathbf{w}$.

(b) Compute $A^5\mathbf{v}$.

(c) Compute $A^5\mathbf{w}$.

(d) Compute $A^5\mathbf{u}$.

## Problem 695

Consider the following system of linear equations:
\begin{align*}
ax_1+bx_2 &=c\\
dx_1+ex_2 &=f\\
gx_1+hx_2 &=i.
\end{align*}

(a) Write down the augmented matrix.

(b) Suppose that the augmented matrix is row equivalent to the identity matrix. Is the system consistent? Justify your answer.

## Problem 694

Let $A, B, C$ be $n\times n$ invertible matrices. When you simplify the expression
$C^{-1}(AB^{-1})^{-1}(CA^{-1})^{-1}C^2,$ which matrix do you get?
(a) $A$
(b) $C^{-1}A^{-1}BC^{-1}AC^2$
(c) $B$
(d) $C^2$
(e) $C^{-1}BC$
(f) $C$

## Problem 693

Let
$A=\begin{bmatrix} -5 & 0 & 1 & 2 \\ 3 &8 & -3 & 7 \\ 0 & 11 & 13 & 28 \end{bmatrix}.$

(a) What is the size of the matrix $A$?
(b) What is the third column of $A$?
(c) Let $a_{ij}$ be the $(i,j)$-entry of $A$. Calculate $a_{23}-a_{31}$.

## Problem 692

Let $A=\begin{bmatrix} 1 & 0 & 3 & -2 \\ 0 &3 & 1 & 1 \\ 1 & 3 & 4 & -1 \end{bmatrix}$. For each of the following vectors, determine whether the vector is in the nullspace $\calN(A)$.

(a) $\begin{bmatrix} -3 \\ 0 \\ 1 \\ 0 \end{bmatrix}$

(b) $\begin{bmatrix} -4 \\ -1 \\ 2 \\ 1 \end{bmatrix}$

(c) $\begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$

(d) $\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$

Then, describe the nullspace $\calN(A)$ of the matrix $A$.

## Problem 691

In this problem, we use the following vectors in $\R^2$.
$\mathbf{a}=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{b}=\begin{bmatrix} 1 \\ 1 \end{bmatrix}, \mathbf{c}=\begin{bmatrix} 2 \\ 3 \end{bmatrix}, \mathbf{d}=\begin{bmatrix} 3 \\ 2 \end{bmatrix}, \mathbf{e}=\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \mathbf{f}=\begin{bmatrix} 5 \\ 6 \end{bmatrix}.$ For each set $S$, determine whether $\Span(S)=\R^2$. If $\Span(S)\neq \R^2$, then give algebraic description for $\Span(S)$ and explain the geometric shape of $\Span(S)$.

(a) $S=\{\mathbf{a}, \mathbf{b}\}$
(b) $S=\{\mathbf{a}, \mathbf{c}\}$
(c) $S=\{\mathbf{c}, \mathbf{d}\}$
(d) $S=\{\mathbf{a}, \mathbf{f}\}$
(e) $S=\{\mathbf{e}, \mathbf{f}\}$
(f) $S=\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}$
(g) $S=\{\mathbf{e}\}$

## Problem 690

Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by
$T( ax^2 + bx + c ) = 2ax + b .$

Is $T$ diagonalizable? If so, find a diagonal matrix which represents $T$. If not, explain why not.

## Problem 689

For this problem, use the complex vectors
$\mathbf{w}_1 = \begin{bmatrix} 1 + i \\ 1 – i \\ 0 \end{bmatrix} , \, \mathbf{w}_2 = \begin{bmatrix} -i \\ 0 \\ 2 – i \end{bmatrix} , \, \mathbf{w}_3 = \begin{bmatrix} 2+i \\ 1 – 3i \\ 2i \end{bmatrix} .$

Suppose $\mathbf{w}_4$ is another complex vector which is orthogonal to both $\mathbf{w}_2$ and $\mathbf{w}_3$, and satisfies $\mathbf{w}_1 \cdot \mathbf{w}_4 = 2i$ and $\| \mathbf{w}_4 \| = 3$.

Calculate the following expressions:

(a) $\mathbf{w}_1 \cdot \mathbf{w}_2$.

(b) $\mathbf{w}_1 \cdot \mathbf{w}_3$.

(c) $((2+i)\mathbf{w}_1 – (1+i)\mathbf{w}_2 ) \cdot \mathbf{w}_4$.

(d) $\| \mathbf{w}_1 \| , \| \mathbf{w}_2 \|$, and $\| \mathbf{w}_3 \|$.

(e) $\| 3 \mathbf{w}_4 \|$.

(f) What is the distance between $\mathbf{w}_2$ and $\mathbf{w}_3$?

## Problem 688

Let $A$ be a $3\times 3$ matrix and let
$\mathbf{v}=\begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} \text{ and } \mathbf{w}=\begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}.$ Suppose that $A\mathbf{v}=-\mathbf{v}$ and $A\mathbf{w}=2\mathbf{w}$.
Then find the vector
$A^5\begin{bmatrix} -1 \\ 8 \\ -9 \end{bmatrix}.$

## Problem 687

For this problem, use the real vectors
$\mathbf{v}_1 = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix} , \mathbf{v}_2 = \begin{bmatrix} 0 \\ 2 \\ -3 \end{bmatrix} , \mathbf{v}_3 = \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} .$ Suppose that $\mathbf{v}_4$ is another vector which is orthogonal to $\mathbf{v}_1$ and $\mathbf{v}_3$, and satisfying
$\mathbf{v}_2 \cdot \mathbf{v}_4 = -3 .$

Calculate the following expressions:

(a) $\mathbf{v}_1 \cdot \mathbf{v}_2$.

(b) $\mathbf{v}_3 \cdot \mathbf{v}_4$.

(c) $( 2 \mathbf{v}_1 + 3 \mathbf{v}_2 – \mathbf{v}_3 ) \cdot \mathbf{v}_4$.

(d) $\| \mathbf{v}_1 \| , \, \| \mathbf{v}_2 \| , \, \| \mathbf{v}_3 \|$.

(e) What is the distance between $\mathbf{v}_1$ and $\mathbf{v}_2$?