## Problem 168

Let
$A=\begin{bmatrix} 1 & -x & 0 & 0 \\ 0 &1 & -x & 0 \\ 0 & 0 & 1 & -x \\ 0 & 1 & 0 & -1 \end{bmatrix}$ be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.

## Problem 167

Let $G, G’$ be groups. Suppose that we have a surjective group homomorphism $f:G\to G’$.
Show that if $G$ is an abelian group, then so is $G’$.

## Problem 166

Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.)

(a) Prove that the subset $W$ is a subspace of $V$.

(b) Find the dimension of $W$.

(The Ohio State University Linear Algebra Exam Problem)

## Problem 165

Let $P_2$ be the vector space of all polynomials of degree two or less.
Consider the subset in $P_2$
$Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},$ where
\begin{align*}
&p_1(x)=1, &p_2(x)=x^2+x+1, \\
&p_3(x)=2x^2, &p_4(x)=x^2-x+1.
\end{align*}

(a) Use the basis $B=\{1, x, x^2\}$ of $P_2$, give the coordinate vectors of the vectors in $Q$.

(b) Find a basis of the span $\Span(Q)$ consisting of vectors in $Q$.

(c) For each vector in $Q$ which is not a basis vector you obtained in (b), express the vector as a linear combination of basis vectors.

(The Ohio State University Linear Algebra Exam Problem)

## Problem 164

Let $T:\R^4 \to \R^3$ be a linear transformation defined by
$T\left (\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \,\right) = \begin{bmatrix} x_1+2x_2+3x_3-x_4 \\ 3x_1+5x_2+8x_3-2x_4 \\ x_1+x_2+2x_3 \end{bmatrix}.$

(a) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$.

(b) Find a basis for the null space of $T$.

(c) Find the rank of the linear transformation $T$.

(The Ohio State University Linear Algebra Exam Problem)

## Problem 163

Let $\Z$ be the additive group of integers. Let $f: \Z \to \Z$ be a group homomorphism.
Then show that there exists an integer $a$ such that
$f(n)=an$ for any integer $n$.

## Problem 162

Let $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$ are vectors in $\R^n$. Suppose that vectors $\mathbf{u}_1$, $\mathbf{u}_2$ are orthogonal and the norm of $\mathbf{u}_2$ is $4$ and $\mathbf{u}_2^{\trans}\mathbf{u}_3=7$. Find the value of the real number $a$ in $\mathbf{u_1}=\mathbf{u_2}+a\mathbf{u}_3$.

(The Ohio State University, Linear Algebra Exam Problem)

## Problem 161

Let $f: H \to G$ be a surjective group homomorphism from a group $H$ to a group $G$.
Let $N$ be a normal subgroup of $H$. Show that the image $f(N)$ is normal in $G$.

## Problem 160

Let $G$ be a finite group and let $H$ be a subset of $G$ such that for any $a,b \in H$, $ab\in H$.

Then show that $H$ is a subgroup of $G$.

## Problem 159

Let $T: \R^2 \to \R^2$ be a linear transformation.
Let
$\mathbf{u}=\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \mathbf{v}=\begin{bmatrix} 3 \\ 5 \end{bmatrix}$ be 2-dimensional vectors.
Suppose that
\begin{align*}
T(\mathbf{u})&=T\left( \begin{bmatrix}
1 \\
2
\end{bmatrix} \right)=\begin{bmatrix}
-3 \\
5
\end{bmatrix},\\
T(\mathbf{v})&=T\left(\begin{bmatrix}
3 \\
5
\end{bmatrix}\right)=\begin{bmatrix}
7 \\
1
\end{bmatrix}.
\end{align*}
Let $\mathbf{w}=\begin{bmatrix} x \\ y \end{bmatrix}\in \R^2$.
Find the formula for $T(\mathbf{w})$ in terms of $x$ and $y$.

## Problem 158

Let $C[3, 10]$ be the vector space consisting of all continuous functions defined on the interval $[3, 10]$. Consider the set
$S=\{ \sqrt{x}, x^2 \}$ in $C[3,10]$.

Show that the set $S$ is linearly independent in $C[3,10]$.

## Problem 157

Let $P_2$ be the vector space of all polynomials of degree two or less.
Consider the subset in $P_2$
$Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},$ where
\begin{align*}
&p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\
&p_3(x)=2x^2, &p_4(x)=2x^2+x+1.
\end{align*}

(a) Use the basis $B=\{1, x, x^2\}$ of $P_2$, give the coordinate vectors of the vectors in $Q$.

(b) Find a basis of the span $\Span(Q)$ consisting of vectors in $Q$.

(c) For each vector in $Q$ which is not a basis vector you obtained in (b), express the vector as a linear combination of basis vectors.

## Problem 156

Let $T: \R^3 \to \R^2$ be a linear transformation such that
$T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 4 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 2 \\ 5 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 3 \\ 6 \end{bmatrix},$ where
$\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \mathbf{e}_3=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$ are the standard unit basis vectors of $\R^3$.
For any vector $\mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\in \R^3$, find a formula for $T(\mathbf{x})$.

## Problem 155

Let $A$ be an $m \times n$ matrix.
Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$.
Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$.

Then find $A\mathbf{w}$.

## Problem 154

Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\right )=\begin{bmatrix} x_1-x_2 \\ x_1+x_2 \\ x_2 \end{bmatrix}$.

(a) Show that $T$ is a linear transformation.

(b) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for each $\mathbf{x} \in \R^2$.

(c) Describe the null space (kernel) and the range of $T$ and give the rank and the nullity of $T$.

## Problem 153

Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient.
Let $W$ be the following subset of $P_3$.
$W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.$ Here $p'(x)$ is the first derivative of $p(x)$ and $p^{\prime\prime}(x)$ is the second derivative of $p(x)$.

Show that $W$ is a subspace of $P_3$ and find a basis for $W$.

## Problem 152

Let $V$ be the vector space of all $2\times 2$ matrices, and let the subset $S$ of $V$ be defined by $S=\{A_1, A_2, A_3, A_4\}$, where
\begin{align*}
A_1=\begin{bmatrix}
1 & 2 \\
-1 & 3
A_2=\begin{bmatrix}
0 & -1 \\
1 & 4
A_3=\begin{bmatrix}
-1 & 0 \\
1 & -10
A_4=\begin{bmatrix}
3 & 7 \\
-2 & 6
\end{bmatrix}.
\end{align*}
Find a basis of the span $\Span(S)$ consisting of vectors in $S$ and find the dimension of $\Span(S)$.

## Problem 151

Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as
$\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,$ where $c_1, c_2, c_3$ are scalars.

## Problem 150

Show that the set
$S=\{1, 1-x, 3+4x+x^2\}$ is a basis of the vector space $P_2$ of all polynomials of degree $2$ or less.

## Problem 149

Let $G$ be a non-abelian simple group. Let $D(G)=[G,G]$ be the commutator subgroup of $G$. Show that $G=D(G)$.