# Category: Linear Algebra

## Problem 356

(a) Let $S=\{\mathbf{v}_1, \mathbf{v}_2\}$ be the set of the following vectors in $\R^4$.
$\mathbf{v}_1=\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}.$ Find an orthogonal basis of the subspace $\Span(S)$ of $\R^4$.

(b) Let $T:\R^2 \to \R^3$ be a linear transformation such that
$T(\mathbf{e}_1)=\mathbf{u}_1 \text{ and } T(\mathbf{e}_2)=\mathbf{u}_2,$ where $\{\mathbf{e}_1, \mathbf{e}_2\}$ is the standard unit vectors of $\R^2$ and
$\mathbf{u}_1=\begin{bmatrix} 5 \\ 1 \\ 2 \end{bmatrix} \text{ and } \mathbf{u}_2=\begin{bmatrix} 8 \\ 2 \\ 6 \end{bmatrix}.$ Then find
$T\left(\, \begin{bmatrix} 3 \\ -2 \end{bmatrix} \,\right).$

## Problem 355

Let $\mathbf{a}, \mathbf{b}$ be vectors in $\R^n$.

Prove the Cauchy-Schwarz inequality:
$|\mathbf{a}\cdot \mathbf{b}|\leq \|\mathbf{a}\|\,\|\mathbf{b}\|.$

## Problem 353

Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying
$T\left(\, \begin{bmatrix} 1 \\ 2 \end{bmatrix}\,\right)=\begin{bmatrix} 3 \\ 4 \\ 5 \end{bmatrix} \text{ and } T\left(\, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}.$ Find a general formula for
$T\left(\, \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \,\right).$

(The Ohio State University, Linear Algebra Math 2568 Exam Problem)

## 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 350

Let $V$ be a vector space over $\R$ and let $B$ be a basis of $V$.
Let $S=\{v_1, v_2, v_3\}$ be a set of vectors in $V$. If the coordinate vectors of these vectors with respect to the basis $B$ is given as follows, then find the dimension of $V$ and the dimension of the span of $S$.
$[v_1]_B=\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, [v_2]_B=\begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, [v_3]_B=\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}.$

## 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 348

Let $A$ be an $n\times n$ complex matrix.
Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as
$p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$ where $a_i$ are real numbers.

Let $C$ be the companion matrix of the polynomial $p(x)$ given by
$C=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \dots & 1 & -a_{n-1} \end{bmatrix}= [\mathbf{e}_2, \mathbf{e}_3, \dots, \mathbf{e}_n, -\mathbf{a}],$ where $\mathbf{e}_i$ is the unit vector in $\C^n$ whose $i$-th entry is $1$ and zero elsewhere, and the vector $\mathbf{a}$ is defined by
$\mathbf{a}=\begin{bmatrix} a_0 \\ a_1 \\ \vdots \\ a_{n-1} \end{bmatrix}.$

Then prove that the following two statements are equivalent.

1. There exists a vector $\mathbf{v}\in \C^n$ such that
$\mathbf{v}, A\mathbf{v}, A^2\mathbf{v}, \dots, A^{n-1}\mathbf{v}$ form a basis of $\C^n$.
2. There exists an invertible matrix $S$ such that $S^{-1}AS=C$.
(Namely, $A$ is similar to the companion matrix of its characteristic polynomial.)

## Problem 347

Let $V$ be a vector space over a scalar field $K$.
Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ be the set of vectors in $V$, where $n \geq 2$.

Then prove that the set $S$ is linearly dependent if and only if at least one of the vectors in $S$ can be written as a linear combination of remaining vectors in $S$.

## Problem 339

Let $\{\mathbf{v}_1, \mathbf{v}_2\}$ be a basis of the vector space $\R^2$, where
$\mathbf{v}_1=\begin{bmatrix} 1 \\ 1 \end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix} 1 \\ -1 \end{bmatrix}.$ The action of a linear transformation $T:\R^2\to \R^3$ on the basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ is given by
\begin{align*}
T(\mathbf{v}_1)=\begin{bmatrix}
2 \\
4 \\
6
\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
0 \\
8 \\
10
\end{bmatrix}.
\end{align*}

Find the formula of $T(\mathbf{x})$, where
$\mathbf{x}=\begin{bmatrix} x \\ y \end{bmatrix}\in \R^2.$

## 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 337

Let $A, B$ be complex $2\times 2$ matrices satisfying the relation
$A=AB-BA.$

Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.

## Problem 336

A complex square ($n\times n$) matrix $A$ is called normal if
$A^* A=A A^*,$ where $A^*$ denotes the conjugate transpose of $A$, that is $A^*=\bar{A}^{\trans}$.
A matrix $A$ is said to be nilpotent if there exists a positive integer $k$ such that $A^k$ is the zero matrix.

(a) Prove that if $A$ is both normal and nilpotent, then $A$ is the zero matrix.
You may use the fact that every normal matrix is diagonalizable.

(b) Give a proof of (a) without referring to eigenvalues and diagonalization.

(c) Let $A, B$ be $n\times n$ complex matrices. Prove that if $A$ is normal and $B$ is nilpotent such that $A+B=I$, then $A=I$, where $I$ is the $n\times n$ identity matrix.

## Problem 330

Let $V$ be the vector space of all $n\times n$ real matrices.
Let us fix a matrix $A\in V$.
Define a map $T: V\to V$ by
$T(X)=AX-XA$ for each $X\in V$.

(a) Prove that $T:V\to V$ is a linear transformation.

(b) Let $B$ be a basis of $V$. Let $P$ be the matrix representation of $T$ with respect to $B$. Find the determinant of $P$.

## Problem 329

Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation.
Prove the followings.

(a) The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$.

(b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the nullspace $\calN(T)$ of $T$.
Let $\mathbf{w}$ be the $n$-dimensional vector that is not in $\calN(T)$. Then
$B’=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}, \mathbf{w}\}$ is a basis of $\R^n$.

(c) Each vector $\mathbf{u}\in \R^n$ can be expressed as
$\mathbf{u}=\mathbf{v}+\frac{T(\mathbf{u})}{T(\mathbf{w})}\mathbf{w}$ for some vector $\mathbf{v}\in \calN(T)$.

## 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 324

Let $T$ be the linear transformation from the $3$-dimensional vector space $\R^3$ to $\R^3$ itself satisfying the following relations.
\begin{align*}
T\left(\, \begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix} \,\right)
=\begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}, \qquad T\left(\, \begin{bmatrix}
2 \\
3 \\
5
\end{bmatrix} \, \right) =
\begin{bmatrix}
0 \\
2 \\
-1
T \left( \, \begin{bmatrix}
0 \\
1 \\
2
\end{bmatrix} \, \right)=
\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}.
\end{align*}
Then for any vector
$\mathbf{x}=\begin{bmatrix} x \\ y \\ z \end{bmatrix}\in \R^3,$ find the formula for $T(\mathbf{x})$.

## 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).