## Problem 283

Let $F$ be a field and let
$H(F)=\left\{\, \begin{bmatrix} 1 & a & b \\ 0 &1 &c \\ 0 & 0 & 1 \end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}$ be the Heisenberg group over $F$.
(The group operation of the Heisenberg group is matrix multiplication.)

Determine which matrices lie in the center of $H(F)$ and prove that the center $Z\big(H(F)\big)$ is isomorphic to the additive group $F$.

## Problem 282

Let $V$ be a finite dimensional vector space over a field $k$ and let $V^*=\Hom(V, k)$ be the dual vector space of $V$.
Let $\{v_i\}_{i=1}^n$ be a basis of $V$ and let $\{v^i\}_{i=1}^n$ be the dual basis of $V^*$. Then prove that
$x=\sum_{i=1}^nv^i(x)v_i$ for any vector $x\in V$.

## Problem 281

(a) For what value(s) of $a$ is the following set $S$ linearly dependent?
$S=\left \{\,\begin{bmatrix} 1 \\ 2 \\ 3 \\ a \end{bmatrix}, \begin{bmatrix} a \\ 0 \\ -1 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ a^2 \\ 7 \end{bmatrix}, \begin{bmatrix} 1 \\ a \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ -2 \\ 3 \\ a^3 \end{bmatrix} \, \right\}.$

(b) Let $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of nonzero vectors in $\R^m$ such that the dot product
$\mathbf{v}_i\cdot \mathbf{v}_j=0$ when $i\neq j$.
Prove that the set is linearly independent.

## Problem 280

Determine whether there exists a nonsingular matrix $A$ if
$A^2=AB+2A,$ where $B$ is the following matrix.
If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.

(a) $B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 1 & 2 & -2 \end{bmatrix}$

(b) $B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.$

## Problem 279

Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent.
\begin{align*}
S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\},
\end{align*}
where
$\mathbf{v}_1=\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ a \\ 4 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 2 \\ b \end{bmatrix}.$

## Problem 278

Prove that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.

## Problem 277

Determine whether the following set of vectors is linearly independent or linearly dependent. If the set is linearly dependent, express one vector in the set as a linear combination of the others.
$\left\{\, \begin{bmatrix} 1 \\ 0 \\ -1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}, \begin{bmatrix} -1 \\ -2 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} -2 \\ -2 \\ 7 \\ 11 \end{bmatrix}\, \right\}.$

## Problem 276

Let $V$ be the vector space of all $2\times 2$ real matrices and let $P_3$ be the vector space of all polynomials of degree $3$ or less with real coefficients.
Let $T: P_3 \to V$ be the linear transformation defined by
$T(a_0+a_1x+a_2x^2+a_3x^3)=\begin{bmatrix} a_0+a_2 & -a_0+a_3\\ a_1-a_2 & -a_1-a_3 \end{bmatrix}$ for any polynomial $a_0+a_1x+a_2x^2+a_3 \in P_3$.
Find a basis for the range of $T$, $\calR(T)$, and determine the rank of $T$, $\rk(T)$, and the nullity of $T$, $\nullity(T)$.
Also, prove that $T$ is not injective.

## Problem 275

Let $A$ be the following $3\times 3$ upper triangular matrix.
$A=\begin{bmatrix} 1 & x & y \\ 0 &1 &z \\ 0 & 0 & 1 \end{bmatrix},$ where $x, y, z$ are some real numbers.

Determine whether the matrix $A$ is invertible or not. If it is invertible, then find the inverse matrix $A^{-1}$.

## Problem 274

Let $U$ and $V$ be subspaces of the vector space $\R^n$.
If neither $U$ nor $V$ is a subset of the other, then prove that the union $U \cup V$ is not a subspace of $\R^n$.

## Problem 273

(a) The given matrix is the augmented matrix for a system of linear equations.
Give the vector form for the general solution.
$\left[\begin{array}{rrrrr|r} 1 & 0 & -1 & 0 &-2 & 0 \\ 0 & 1 & 2 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ \end{array} \right].$

(b) Let
$A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}, C=\begin{bmatrix} 1 & 2\\ 0& 6 \end{bmatrix}, \mathbf{v}=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}.$ Then compute and simplify the following expression.
$\mathbf{v}^{\trans}\left( A^{\trans}-(A-B)^{\trans}\right)C.$

## Problem 272

Let
$A=\begin{bmatrix} 1 & 3\\ 2& 4 \end{bmatrix}.$ Then

(a) Find all matrices
$B=\begin{bmatrix} x & y\\ z& w \end{bmatrix}$ such that $AB=BA$.

(b) Use the results of part (a) to exhibit $2\times 2$ matrices $B$ and $C$ such that
$AB=BA \text{ and } AC \neq CA.$

## Problem 271

Let $A$ be an $n\times n$ singular matrix.
Then prove that there exists a nonzero $n\times n$ matrix $B$ such that
$AB=O,$ where $O$ is the $n\times n$ zero matrix.

## Problem 270

Let
$A=\begin{bmatrix} 4 & 1\\ 3& 2 \end{bmatrix}$ and consider the following subset $V$ of the 2-dimensional vector space $\R^2$.
$V=\{\mathbf{x}\in \R^2 \mid A\mathbf{x}=5\mathbf{x}\}.$

(a) Prove that the subset $V$ is a subspace of $\R^2$.

(b) Find a basis for $V$ and determine the dimension of $V$.

## Problem 269

Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$.
Then prove the following statements.

(a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number.

(b) The rank of $A$ is even.

## Problem 268

Let $G$ be a group with the identity element $e$ and suppose that we have a group homomorphism $\phi$ from the direct product $G \times G$ to $G$ satisfying
$\phi(e, g)=g \text{ and } \phi(g, e)=g, \tag{*}$ for any $g\in G$.

Let $\mu: G\times G \to G$ be a map defined by
$\mu(g, h)=gh.$ (That is, $\mu$ is the group operation on $G$.)

Then prove that $\phi=\mu$.
Also prove that the group $G$ is abelian.

## Problem 267

Solve the following system of linear equations by transforming its augmented matrix to reduced echelon form (Gauss-Jordan elimination).

Find the vector form for the general solution.
\begin{align*}
x_1-x_3-3x_5&=1\\
3x_1+x_2-x_3+x_4-9x_5&=3\\
x_1-x_3+x_4-2x_5&=1.
\end{align*}

## Problem 266

Let $A$ be an $n \times n$ matrix satisfying
$A^2+c_1A+c_0I=O,$ where $c_0, c_1$ are scalars, $I$ is the $n\times n$ identity matrix, and $O$ is the $n\times n$ zero matrix.

Prove that if $c_0\neq 0$, then the matrix $A$ is invertible (nonsingular).
How about the converse? Namely, is it true that if $c_0=0$, then the matrix $A$ is not invertible?

## Problem 265

For a real number $a$, consider $2\times 2$ matrices $A, P, Q$ satisfying the following five conditions.

1. $A=aP+(a+1)Q$
2. $P^2=P$
3. $Q^2=Q$
4. $PQ=O$
5. $QP=O$,

where $O$ is the $2\times 2$ zero matrix.
Then do the following problems.

(a) Prove that $(P+Q)A=A$.

(b) Suppose $a$ is a positive real number and let
$A=\begin{bmatrix} a & 0\\ 1& a+1 \end{bmatrix}.$ Then find all matrices $P, Q$ satisfying conditions (1)-(5).

(c) Let $n$ be an integer greater than $1$. For any integer $k$, $2\leq k \leq n$, we define the matrix
$A_k=\begin{bmatrix} k & 0\\ 1& k+1 \end{bmatrix}.$ Then calculate and simplify the matrix product
$A_nA_{n-1}A_{n-2}\cdots A_2.$

(Tokyo University Entrance Exam 2007)

Let $\Z$ be the ring of integers and let $R$ be a ring with unity.
Determine all the ring homomorphisms from $\Z$ to $R$.