# Tagged: subspace

## Problem 664

Let $V$ be the vector space of $k \times k$ matrices. Then for fixed matrices $R, S \in V$, define the subset $W = \{ R A S \mid A \in V \}$.

Prove that $W$ is a vector subspace of $V$.

## Problem 663

Let $\R^2$ be the $x$-$y$-plane. Then $\R^2$ is a vector space. A line $\ell \subset \mathbb{R}^2$ with slope $m$ and $y$-intercept $b$ is defined by
$\ell = \{ (x, y) \in \mathbb{R}^2 \mid y = mx + b \} .$

Prove that $\ell$ is a subspace of $\mathbb{R}^2$ if and only if $b = 0$.

## Problem 662

For what real values of $a$ is the set
$W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}$ a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?

## Problem 661

Let $C(\mathbb{R})$ be the vector space of real-valued functions on $\mathbb{R}$.

Consider the set of functions $W = \{ f(x) = a + b \cos(x) + c \cos(2x) \mid a, b, c \in \mathbb{R} \}$.

Prove that $W$ is a vector subspace of $C(\mathbb{R})$.

## Problem 660

Let $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define
$W = \{ A \in V \mid AM = MA \}.$ The set $W$ here is called the centralizer of $M$ in $V$.

Prove that $W$ is a subspace of $V$.

## Problem 659

Fix the row vector $\mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}$, and let $\R^3$ be the vector space of $3 \times 1$ column vectors. Define
$W = \{ \mathbf{v} \in \R^3 \mid \mathbf{b} \mathbf{v} = 0 \}.$ Prove that $W$ is a vector subspace of $\R^3$, and find a basis for $W$.

## Problem 658

Let $V$ be the vector space of $n \times n$ matrices with real coefficients, and define
$W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.$ The set $W$ is called the center of $V$.

Prove that $W$ is a subspace of $V$.

## Problem 612

Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$.
Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$.

(a) Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ is a basis for $W$.

(b) Prove that the set $\{\sin^2(x)-\cos^2(x), 1\}$ is a basis for $W$.

## Problem 611

An $n\times n$ matrix $A$ is called orthogonal if $A^{\trans}A=I$.
Let $V$ be the vector space of all real $2\times 2$ matrices.

Consider the subset
$W:=\{A\in V \mid \text{A is an orthogonal matrix}\}.$ Prove or disprove that $W$ is a subspace of $V$.

## Problem 604

Let
$A=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 &1 & 1 & 1 \\ 1 & -1 & 0 & 0 \\ 0 & 2 & 2 & 2\\ 0 & 0 & 0 & 0 \end{bmatrix}.$

(a) Find a basis for the null space $\calN(A)$.

(b) Find a basis of the range $\calR(A)$.

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

(The Ohio State University, Linear Algebra Midterm)

## Problem 602

Let $W$ be a subspace of $\R^4$ with a basis
$\left\{\, \begin{bmatrix} 1 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix} \,\right\}.$

Find an orthonormal basis of $W$.

(The Ohio State University, Linear Algebra Midterm)

## Problem 601

Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers.
Let
$W=\left\{\, A\in V \quad \middle | \quad A=\begin{bmatrix} a & b\\ c& -a \end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.$

(a) Show that $W$ is a subspace of $V$.

(b) Find a basis of $W$.

(c) Find the dimension of $W$.

(The Ohio State University, Linear Algebra Midterm)

## Problem 600

Let $\mathbf{v}_1=\begin{bmatrix} 2/3 \\ 2/3 \\ 1/3 \end{bmatrix}$ be a vector in $\R^3$.

Find an orthonormal basis for $\R^3$ containing the vector $\mathbf{v}_1$.

## Problem 595

Let $U$ and $V$ be subspaces of the $n$-dimensional vector space $\R^n$.

Prove that the intersection $U\cap V$ is also a subspace of $\R^n$.

## Problem 590

Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let
$V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}$ be a subset in $C[-1, 1]$.

(a) Prove that $V$ is a subspace of $C[-1, 1]$.

(b) Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$.

(c) Prove that
$B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}$ is a basis for $V$.

## Problem 586

Let $V$ be the vector space over $\R$ of all real $2\times 2$ matrices.
Let $W$ be the subset of $V$ consisting of all symmetric matrices.

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

(b) Find a basis of $W$.

(c) Determine the dimension of $W$.

## Problem 581

Let $V$ be a vector space over a scalar field $K$.
Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset
$W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } a_1+a_2+\cdots+a_k=0\}.$ So each element of $W$ is a linear combination of vectors $\mathbf{v}_1, \dots, \mathbf{v}_k$ such that the sum of the coefficients is zero.

Prove that $W$ is a subspace of $V$.

## Problem 578

Let $V$ be a subset of $\R^4$ consisting of vectors that are perpendicular to vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, where
$\mathbf{a}=\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \quad \mathbf{b}=\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \quad \mathbf{c}=\begin{bmatrix} 0 \\ 1 \\ -1 \\ 0 \end{bmatrix}.$

Namely,
$V=\{\mathbf{x}\in \R^4 \mid \mathbf{a}^{\trans}\mathbf{x}=0, \mathbf{b}^{\trans}\mathbf{x}=0, \text{ and } \mathbf{c}^{\trans}\mathbf{x}=0\}.$

(a) Prove that $V$ is a subspace of $\R^4$.

(b) Find a basis of $V$.

(c) Determine the dimension of $V$.

## Problem 577

Let $V$ be a subspace of $\R^n$.
Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ is a basis of the subspace $V$.

Prove that every basis of $V$ consists of $k$ vectors in $V$.

## Problem 576

Let $V$ be a subspace of $\R^n$.
Suppose that
$S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}$ is a spanning set for $V$.

Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.