# Category: Linear Algebra

## Problem 85

Consider a polynomial
$p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$ where $a_i$ are real numbers.
Define the matrix
$A=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \dots & 1 & -a_{n-1} \end{bmatrix}.$

Then prove that the characteristic polynomial $\det(xI-A)$ of $A$ is the polynomial $p(x)$.
The matrix is called the companion matrix of the polynomial $p(x)$.

## Problem 84

Let $A$ be an $n \times n$ complex matrix such that $A^k=I$, where $I$ is the $n \times n$ identity matrix.

Show that the matrix $A$ is diagonalizable.

## Problem 80

Let $V$ be a finite dimensional vector space over a field $K$ and let $\End (V)$ be the vector space of linear transformations from $V$ to $V$.
Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$ be a basis for $V$.
Show that the map $\phi:\End (V) \to V^{\oplus n}$ defined by $f\mapsto (f(\mathbf{v}_1), \dots, f(\mathbf{v}_n))$ is an isomorphism.
Here $V^{\oplus n}=V\oplus \dots \oplus V$, the direct sum of $n$ copies of $V$.

## Problem 79

Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero.
That is,

\begin{equation*}
V:=\left\{ A=\begin{bmatrix}
a_{11} & 0 & \dots & 0 \\
0 &a_{22} & \dots & 0 \\
0 & 0 & \ddots & \vdots \\
0 & 0 & \dots & a_{nn}
\begin{array}{l}
a_{11}, \dots, a_{nn} \in \C,\\
\tr(A)=0 \\
\end{array}
\right\}
\end{equation*}

Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.

(a) Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)

(b) Show that matrices
$E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}$ are a basis for the vector space $V$.

(c) Find the dimension of $V$.

## Problem 78

Determine whether the following sentence is True or False.

(Purdue University Linear Algebra Exam)

## Problem 77

A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix.

(a) If $A$ is a nilpotent $n \times n$ matrix and $B$ is an $n\times n$ matrix such that $AB=BA$. Show that the product $AB$ is nilpotent.

(b) Let $P$ be an invertible $n \times n$ matrix and let $N$ be a nilpotent $n\times n$ matrix. Is the product $PN$ nilpotent? If so, prove it. If not, give a counterexample.

## Problem 76

Let A be the matrix
$\begin{bmatrix} 1 & -1 & 0 \\ 0 &1 &-1 \\ 0 & 0 & 1 \end{bmatrix}.$ Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse.

(The Ohio State University Linear Algebra Exam)

## Problem 75

Let $\Q$ denote the set of rational numbers (i.e., fractions of integers). Let $V$ denote the set of the form $x+y \sqrt{2}$ where $x,y \in \Q$. You may take for granted that the set $V$ is a vector space over the field $\Q$.

(a) Show that $B=\{1, \sqrt{2}\}$ is a basis for the vector space $V$ over $\Q$.

(b) Let $\alpha=a+b\sqrt{2} \in V$, and let $T_{\alpha}: V \to V$ be the map defined by
$T_{\alpha}(x+y\sqrt{2}):=(ax+2by)+(ay+bx)\sqrt{2}\in V$ for any $x+y\sqrt{2} \in V$.
Show that $T_{\alpha}$ is a linear transformation.

(c) Let $\begin{bmatrix} x \\ y \end{bmatrix}_B=x+y \sqrt{2}$.
Find the matrix $T_B$ such that
$T_{\alpha} (x+y \sqrt{2})=\left( T_B\begin{bmatrix} x \\ y \end{bmatrix}\right)_B,$ and compute $\det T_B$.

(The Ohio State University, Linear Algebra Exam)

## Problem 73

Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers.

Show that exponential functions
$e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}$ are linearly independent over $\R$.

## Problem 72

(a) Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that the entries of the matrix $A$ satisfy the following relation.
$|a_{ii}|>|a_{i1}|+\cdots +|a_{i\,i-1}|+|a_{i \, i+1}|+\cdots +|a_{in}|$ for all $1 \leq i \leq n$.
Show that the matrix $A$ is nonsingular.

(b) Let $B=(b_{ij})$ be an $n \times n$ matrix whose entries satisfy the relation
$|b_{i\,i}|=1 \hspace{0.5cm} \text{ and }\hspace{0.5cm} |b_{ij}|<\frac{1}{n-1}$ for all $i$ and $j$ with $i \neq j$.
Prove that the matrix $B$ is nonsingular.

(c)
Determine whether the following matrix is nonsingular or not.
$C=\begin{bmatrix} \pi & e & e^2/2\pi^2 \\[5 pt] e^2/2\pi^2 &\pi &e \\[5pt] e & e^2/2\pi^2 & \pi \end{bmatrix},$ where $\pi=3.14159\dots$, and $e=2.71828\dots$ is Euler’s number (or Napier’s constant).

## Problem 71

Let $P_2(\R)$ be the vector space over $\R$ consisting of all polynomials with real coefficients of degree $2$ or less.
Let $B=\{1,x,x^2\}$ be a basis of the vector space $P_2(\R)$.
For each linear transformation $T:P_2(\R) \to P_2(\R)$ defined below, find the matrix representation of $T$ with respect to the basis $B$. For $f(x)\in P_2(\R)$, define $T$ as follows.

(a) $T(f(x))=\frac{\mathrm{d}^2}{\mathrm{d}x^2} f(x)-3\frac{\mathrm{d}}{\mathrm{d}x}f(x)$

(b) $T(f(x))=\int_{-1}^1\! (t-x)^2f(t) \,\mathrm{d}t$

(c) $T(f(x))=e^x \frac{\mathrm{d}}{\mathrm{d}x}(e^{-x}f(x))$

## Problem 70

Suppose that $A$ is an $n \times n$ matrix with eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$.

(a) If $A$ is invertible, is $\mathbf{v}$ an eigenvector of $A^{-1}$? If so, what is the corresponding eigenvalue? If not, explain why not.

(b) Is $3\mathbf{v}$ an eigenvector of $A$? If so, what is the corresponding eigenvalue? If not, explain why not.

(Stanford University, Linear Algebra Exam)

## Problem 69

Let $F$ and $H$ be an $n\times n$ matrices satisfying the relation
$HF-FH=-2F.$

(a) Find the trace of the matrix $F$.

(b) Let $\lambda$ be an eigenvalue of $H$ and let $\mathbf{v}$ be an eigenvector corresponding to $\lambda$. Show that there exists an positive integer $N$ such that $F^N\mathbf{v}=\mathbf{0}$.

## Problem 68

Let $H$ and $E$ be $n \times n$ matrices satisfying the relation
$HE-EH=2E.$ Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$.
Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then prove that
$E\mathbf{x}=\mathbf{0}.$

## Problem 67

Answer the following questions regarding eigenvalues of a real matrix.

(a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers.
(b) Find the eigenvalues of the matrix
$B=\begin{bmatrix} -2 & -1\\ 5& 2 \end{bmatrix}.$

(The Ohio State University, Linear Algebra Exam)

## Problem 66

Consider the matrix
$A=\begin{bmatrix} 1 & 2 & 1 \\ 2 &5 &4 \\ 1 & 1 & 0 \end{bmatrix}.$

(a) Calculate the inverse matrix $A^{-1}$. If you think the matrix $A$ is not invertible, then explain why.

(b) Are the vectors
$\mathbf{A}_1=\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \mathbf{A}_2=\begin{bmatrix} 2 \\ 5 \\ 1 \end{bmatrix}, \text{ and } \mathbf{A}_3=\begin{bmatrix} 1 \\ 4 \\ 0 \end{bmatrix}$ linearly independent?

(c) Write the vector $\mathbf{b}=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ as a linear combination of $\mathbf{A}_1$, $\mathbf{A}_2$, and $\mathbf{A}_3$.

(The Ohio State University, Linear Algebra Exam)

## Problem 65

Consider the system of linear equations
\begin{align*}
x_1&= 2, \\
-2x_1 + x_2 &= 3, \\
5x_1-4x_2 +x_3 &= 2
\end{align*}

(a) Find the coefficient matrix and its inverse matrix.

(b) Using the inverse matrix, solve the system of linear equations.

(The Ohio State University, Linear Algebra Exam)

## Problem 63

Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers.

Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$.

## Problem 62

Let $T: \R^n \to \R^m$ be a linear transformation.
Suppose that $S=\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a subset of $\R^n$ such that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$.

Prove that the set $S$ is linearly independent.

## Problem 61

Let $V$ and $W$ be subspaces of $\R^n$ such that $V \cap W =\{\mathbf{0}\}$ and $\dim(V)+\dim(W)=n$.

(a) If $\mathbf{v}+\mathbf{w}=\mathbf{0}$, where $\mathbf{v}\in V$ and $\mathbf{w}\in W$, then show that $\mathbf{v}=\mathbf{0}$ and $\mathbf{w}=\mathbf{0}$.

(b) If $B_1$ is a basis for the subspace $V$ and $B_2$ is a basis for the subspace $W$, then show that the union $B_1\cup B_2$ is a basis for $R^n$.

(c) If $\mathbf{x}$ is in $\R^n$, then show that $\mathbf{x}$ can be written in the form $\mathbf{x}=\mathbf{v}+\mathbf{w}$, where $\mathbf{v}\in V$ and $\mathbf{w} \in W$.

(d) Show that the representation obtained in part (c) is unique.