# Tagged: subspace

## Problem 714

Let $W$ be the set of $3\times 3$ skew-symmetric matrices. Show that $W$ is a subspace of the vector space $V$ of all $3\times 3$ matrices. Then, exhibit a spanning set for $W$.

## Problem 712

Let $A$ be an $m \times n$ matrix.
Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$.

## Problem 710

Find a basis for $\Span(S)$ where $S= \left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} , \begin{bmatrix} -1 \\ -2 \\ -1 \end{bmatrix} , \begin{bmatrix} 2 \\ 6 \\ -2 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix} \right\}$.

## Problem 709

Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where
$\mathbf{v}_{1}= \begin{bmatrix} 1 \\ 2 \\ 2 \\ -1 \end{bmatrix} ,\;\mathbf{v}_{2}= \begin{bmatrix} 1 \\ 3 \\ 1 \\ 1 \end{bmatrix} ,\;\mathbf{v}_{3}= \begin{bmatrix} 1 \\ 5 \\ -1 \\ 5 \end{bmatrix} ,\;\mathbf{v}_{4}= \begin{bmatrix} 1 \\ 1 \\ 4 \\ -1 \end{bmatrix} ,\;\mathbf{v}_{5}= \begin{bmatrix} 2 \\ 7 \\ 0 \\ 2 \end{bmatrix} .$ Find a basis for the span $\Span(S)$.

## Problem 707

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 $\R^3$. Is it possible that $S_2=\{\mathbf{v}_1\}$ is a spanning set for $V$?

## 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 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 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 $\R^3$.

## 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)

## Find an Orthonormal Basis of $\R^3$ Containing a Given Vector
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 $\mathbf{v}_1$.