# Tagged: linear algebra

## 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 146

Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix.

Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity matrix.

## Problem 143

Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.

(a) The set $S$ consisting of all $n\times n$ symmetric matrices.

(b) The set $T$ consisting of all $n \times n$ skew-symmetric matrices.

(c) The set $U$ consisting of all $n\times n$ nonsingular matrices.

## Problem 142

Let $T:\R^2 \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{e}_2)=\mathbf{u}_2$, where $\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ are unit vectors of $\R^2$ and
$\mathbf{u}_1= \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix}, \quad \mathbf{u}_2=\begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}.$ Then find $T\left(\begin{bmatrix} 3 \\ -2 \end{bmatrix}\right)$.

## Problem 141

Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$.
Let $\mathbf{u}_{n+1}\in V$. Show that $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n, \mathbf{u}_{n+1}$ are linearly independent if and only if $\mathbf{u}_{n+1} \not \in U$.

## Problem 140

Let $A$ be an $m\times n$ matrix. The nullspace of $A$ is denoted by $\calN(A)$.
The dimension of the nullspace of $A$ is called the nullity of $A$.
Prove the followings.

(a) $\calN(A)=\calN(A^{\trans}A)$.

(b) $\rk(A)=\rk(A^{\trans}A)$.

## Problem 139

Let $A_1, A_2, \dots, A_m$ be $n\times n$ Hermitian matrices. Show that if
$A_1^2+A_2^2+\cdots+A_m^2=\calO,$ where $\calO$ is the $n \times n$ zero matrix, then we have $A_i=\calO$ for each $i=1,2, \dots, m$.

## Problem 138

Find the determinant of the matix
$A=\begin{bmatrix} 100 & 101 & 102 \\ 101 &102 &103 \\ 102 & 103 & 104 \end{bmatrix}.$

## Problem 137

Let $P_n(\R)$ be the vector space over $\R$ consisting of all degree $n$ or less real coefficient polynomials. Let
$U=\{ p(x) \in P_n(\R) \mid p(1)=0\}$ be a subspace of $P_n(\R)$.

Find a basis for $U$ and determine the dimension of $U$.

## Problem 136

Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.

## Problem 135

Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings.

(a) $\rk(AB) \leq \rk(A)$.

(b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$.

## Problem 134

Let $V$ be the vector space over $\R$ of all real valued functions defined on the interval $[0,1]$. Determine whether the following subsets of $V$ are subspaces or not.

(a) $S=\{f(x) \in V \mid f(0)=f(1)\}$.

(b) $T=\{f(x) \in V \mid f(0)=f(1)+3\}$.

## Problem 133

Find a square root of the matrix
$A=\begin{bmatrix} 1 & 3 & -3 \\ 0 &4 &5 \\ 0 & 0 & 9 \end{bmatrix}.$

How many square roots does this matrix have?

(University of California, Berkeley Qualifying Exam)

## Problem 132

Let
$A=\begin{bmatrix} 1 & 1 & 0 \\ 1 &1 &0 \end{bmatrix}$ be a matrix.

Find a basis of the null space of the matrix $A$.

(Remark: a null space is also called a kernel.)

## Problem 131

Let $V$ be the following subspace of the $4$-dimensional vector space $\R^4$.
$V:=\left\{ \quad\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \in \R^4 \quad \middle| \quad x_1-x_2+x_3-x_4=0 \quad\right\}.$ Find a basis of the subspace $V$ and its dimension.

## Problem 126

Let $A$ be the following $3 \times 3$ matrix.
$A=\begin{bmatrix} 1 & 1 & -1 \\ 0 &1 &2 \\ 1 & 1 & a \end{bmatrix}.$ Determine the values of $a$ so that the matrix $A$ is nonsingular.

## Problem 125

Let $S$ be the following subset of the 3-dimensional vector space $\R^3$.
$S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, x_1, x_2, x_3 \in \Z \right\},$ where $\Z$ is the set of all integers.
Determine whether $S$ is a subspace of $\R^3$.

## Problem 121

Let $A$ be an $m \times n$ real matrix. Then the null space $\calN(A)$ of $A$ is defined by
$\calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.$ That is, the null space is the set of solutions to the homogeneous system $A\mathbf{x}=\mathbf{0}_m$.

Prove that the null space $\calN(A)$ is a subspace of the vector space $\R^n$.
(Note that the null space is also called the kernel of $A$.)

## Problem 120

Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r$ are linearly dependent $n$-dimensional real vectors.

For any vector $\mathbf{v}_{r+1} \in \R^n$, determine whether the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r, \mathbf{v}_{r+1}$ are linearly independent or linearly dependent.

## Problem 119

Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by
$W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.$

Prove that the subset $W$ is a subspace of $\R^3$.