# Tagged: subspace

## Problem 366

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

(a) Find an orthonormal basis of the null space of $A$.

(b) Find the rank of $A$.

(c) Find an orthonormal basis of the row space of $A$.

(The Ohio State University, Linear Algebra Exam Problem)

## Problem 364

These are True or False problems.
For each of the following statements, determine if it contains a wrong information or not.

1. Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$.
2. The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because $f(0)=1\neq 0$.
3. Since we have $\sin(x+y)=\sin(x)+\sin(y)$, the function $\sin(x)$ is a linear transformation.
4. The set
$\left\{\, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \,\right\}$ is an orthonormal set.

(Linear Algebra Exam Problem, The Ohio State University)

## Problem 352

A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors
$\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\in \R^n$ satisfying the linear equation of the form
$a_1x_1+a_2x_2+\cdots+a_nx_n=b,$ where $a_1, a_2, \dots, a_n$ (at least one of $a_1, a_2, \dots, a_n$ is nonzero) and $b$ are real numbers.
Here at least one of $a_1, a_2, \dots, a_n$ is nonzero.

Consider the hyperplane $P$ in $\R^n$ described by the linear equation
$a_1x_1+a_2x_2+\cdots+a_nx_n=0,$ where $a_1, a_2, \dots, a_n$ are some fixed real numbers and not all of these are zero.
(The constant term $b$ is zero.)

Then prove that the hyperplane $P$ is a subspace of $R^{n}$ of dimension $n-1$.

## Problem 349

Let $V$ be the vector space of all $2\times 2$ real matrices.
Let $S=\{A_1, A_2, A_3, A_4\}$, where
$A_1=\begin{bmatrix} 1 & 2\\ -1& 3 \end{bmatrix}, A_2=\begin{bmatrix} 0 & -1\\ 1& 4 \end{bmatrix}, A_3=\begin{bmatrix} -1 & 0\\ 1& -10 \end{bmatrix}, A_4=\begin{bmatrix} 3 & 7\\ -2& 6 \end{bmatrix}.$ Then find a basis for the span $\Span(S)$.

## Problem 338

Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.
(1) $S_1=\left \{\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}$ in the vector space $\R^3$.

(2) $S_2=\left \{\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 \quad \middle | \quad x_1-4x_2+5x_3=2 \,\right \}$ in the vector space $\R^3$.

(3) $S_3=\left \{\, \begin{bmatrix} x \\ y \end{bmatrix}\in \R^2 \quad \middle | \quad y=x^2 \quad \,\right \}$ in the vector space $\R^2$.

(4) Let $P_4$ be the vector space of all polynomials of degree $4$ or less with real coefficients.
$S_4=\{ f(x)\in P_4 \mid f(1) \text{ is an integer}\}$ in the vector space $P_4$.

(5) $S_5=\{ f(x)\in P_4 \mid f(1) \text{ is a rational number}\}$ in the vector space $P_4$.

(6) Let $M_{2 \times 2}$ be the vector space of all $2\times 2$ real matrices.
$S_6=\{ A\in M_{2\times 2} \mid \det(A) \neq 0\}$ in the vector space $M_{2\times 2}$.

(7) $S_7=\{ A\in M_{2\times 2} \mid \det(A)=0\}$ in the vector space $M_{2\times 2}$.

(Linear Algebra Exam Problem, the Ohio State University)

(8) Let $C[-1, 1]$ be the vector space of all real continuous functions defined on the interval $[a, b]$.
$S_8=\{ f(x)\in C[-2,2] \mid f(-1)f(1)=0\}$ in the vector space $C[-2, 2]$.

(9) $S_9=\{ f(x) \in C[-1, 1] \mid f(x)\geq 0 \text{ for all } -1\leq x \leq 1\}$ in the vector space $C[-1, 1]$.

(10) Let $C^2[a, b]$ be the vector space of all real-valued functions $f(x)$ defined on $[a, b]$, where $f(x), f'(x)$, and $f^{\prime\prime}(x)$ are continuous on $[a, b]$. Here $f'(x), f^{\prime\prime}(x)$ are the first and second derivative of $f(x)$.
$S_{10}=\{ f(x) \in C^2[-1, 1] \mid f^{\prime\prime}(x)+f(x)=\sin(x) \text{ for all } -1\leq x \leq 1\}$ in the vector space $C[-1, 1]$.

(11) Let $S_{11}$ be the set of real polynomials of degree exactly $k$, where $k \geq 1$ is an integer, in the vector space $P_k$.

(12) Let $V$ be a vector space and $W \subset V$ a vector subspace. Define the subset $S_{12}$ to be the complement of $W$,
$V \setminus W = \{ \mathbf{v} \in V \mid \mathbf{v} \not\in W \}.$

## Problem 328

(a) Let $C[-1,1]$ be the vector space over $\R$ of all real-valued continuous functions defined on the interval $[-1, 1]$.
Consider the subset $F$ of $C[-1, 1]$ defined by
$F=\{ f(x)\in C[-1, 1] \mid f(0) \text{ is an integer}\}.$ Prove or disprove that $F$ is a subspace of $C[-1, 1]$.

(b) Let $n$ be a positive integer.
An $n\times n$ matrix $A$ is called skew-symmetric if $A^{\trans}=-A$.
Let $M_{n\times n}$ be the vector space over $\R$ of all $n\times n$ real matrices.
Consider the subset $W$ of $M_{n\times n}$ defined by
$W=\{A\in M_{n\times n} \mid A \text{ is skew-symmetric}\}.$ Prove or disprove that $W$ is a subspace of $M_{n\times n}$.

## Problem 327

Let $A$ be the matrix for a linear transformation $T:\R^n \to \R^n$ with respect to the standard basis of $\R^n$.
We assume that $A$ is idempotent, that is, $A^2=A$.
Then prove that
$\R^n=\im(T) \oplus \ker(T).$

## Problem 323

Suppose that $A$ is $2\times 2$ matrix that has eigenvalues $-1$ and $3$.
Then for each positive integer $n$ find $a_n$ and $b_n$ such that
$A^{n+1}=a_nA+b_nI,$ where $I$ is the $2\times 2$ identity matrix.

## Problem 321

Let $V$ be a real vector space of all real sequences
$(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).$ Let $U$ be a subspace of $V$ defined by
$U=\{(a_i)_{i=1}^{\infty}\in V \mid a_{n+2}=2a_{n+1}+3a_{n} \text{ for } n=1, 2,\dots \}.$ Let $T$ be the linear transformation from $U$ to $U$ defined by
$T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots).$

(a) Find the eigenvalues and eigenvectors of the linear transformation $T$.

(b) Use the result of (a), find a sequence $(a_i)_{i=1}^{\infty}$ satisfying $a_1=2, a_2=7$.

## Problem 320

(a) Let $A=\begin{bmatrix} 1 & 3 & 0 & 0 \\ 1 &3 & 1 & 2 \\ 1 & 3 & 1 & 2 \end{bmatrix}$.
Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$.

(b) Find the rank and nullity of the matrix $A$ in part (a).

## Problem 313

(a) Let $A=\begin{bmatrix} 1 & 2 & 1 \\ 3 &6 &4 \end{bmatrix}$ and let
$\mathbf{a}=\begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix} 1 \\ 1 \end{bmatrix}.$ For each of the vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$, determine whether the vector is in the null space $\calN(A)$. Do the same for the range $\calR(A)$.

(b) Find a basis of the null space of the matrix $B=\begin{bmatrix} 1 & 1 & 2 \\ -2 &-2 &-4 \end{bmatrix}$.

## Problem 311

Let $A$ and $B$ be $n\times n$ matrices. Then prove that
$\calN(A)\cap \calN(B) \subset \calN(A+B),$ where $\calN(A)$ is the null space (kernel) of the matrix $A$.

## Problem 309

Let $V$ be a real vector space of all real sequences
$(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).$ Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ for $k=1, 2, \dots$.

(a) Let
\begin{align*}
\mathbf{u}_1&=(1, 0, -3, -15, -66, \dots)\\
\mathbf{u}_2&=(0, 1, 5, 22, 95, \dots)
\end{align*}
be vectors in $U$. Prove that $\{\mathbf{u}_1, \mathbf{u}_2\}$ is a basis of $U$ and conclude that the dimension of $U$ is $2$.

(b) Let $T$ be a map from $U$ to $U$ defined by
$T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots).$ Verify that the map $T$ actually sends a vector $(a_i)_{i=1}^{\infty}\in V$ to a vector $T\big((a_i)_{i=1}^{\infty}\big)$ in $U$, and show that $T$ is a linear transformation from $U$ to $U$.

(c) With respect to the basis $\{\mathbf{u}_1, \mathbf{u}_2\}$ obtained in (a), find the matrix representation $A$ of the linear transformation $T:U \to U$ from (b).

## Problem 308

Let $V$ be a real vector space of all real sequences
$(a_i)_{i=1}^{\infty}=(a_1, a_2, \cdots).$ Let $U$ be the subset of $V$ defined by
$U=\{ (a_i)_{i=1}^{\infty} \in V \mid a_{k+2}-5a_{k+1}+3a_{k}=0, k=1, 2, \dots \}.$

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

## Problem 304

Problem 1 Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by
$W=\left\{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.$

(a) Which of the following vectors are in the subset $W$? Choose all vectors that belong to $W$.
$(1) \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \qquad(2) \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} \qquad(3)\begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} \qquad(4) \begin{bmatrix} 0 \\ 0 \end{bmatrix} \qquad(5) \begin{bmatrix} 1 & 2 & 4 \\ 1 &2 &4 \end{bmatrix} \qquad(6) \begin{bmatrix} 1 \\ -1 \\ -2 \end{bmatrix}.$

(b) Determine whether $W$ is a subspace of $\R^3$ or not.

Problem 2 Let $W$ be the subset of $\R^3$ defined by
$W=\left\{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 \quad \middle| \quad x_1=3x_2 \text{ and } x_3=0 \right\}.$ Determine whether the subset $W$ is a subspace of $\R^3$ or not.

## Problem 294

Prove that every plane in the $3$-dimensional space $\R^3$ that passes through the origin is a subspace of $\R^3$.

## Problem 292

Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$.
Then prove that $V$ is a subspace of $\R^n$.

## Problem 287

Let $V$ be the vector space of all $3\times 3$ real matrices.
Let $A$ be the matrix given below and we define
$W=\{M\in V \mid AM=MA\}.$ That is, $W$ consists of matrices that commute with $A$.
Then $W$ is a subspace of $V$.

Determine which matrices are in the subspace $W$ and find the dimension of $W$.

(a) $A=\begin{bmatrix} a & 0 & 0 \\ 0 &b &0 \\ 0 & 0 & c \end{bmatrix},$ where $a, b, c$ are distinct real numbers.

(b) $A=\begin{bmatrix} a & 0 & 0 \\ 0 &a &0 \\ 0 & 0 & b \end{bmatrix},$ where $a, b$ are distinct real numbers.

## Problem 285

Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let
$W=\{ f(x)\in V \mid f(x)=f(1-x) \text{ for } x\in [0,1]\}$ be a subset of $V$. Determine whether the subset $W$ is a subspace of the vector space $V$.

## 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 $\R^n$.