# Category: Linear Algebra

## Problem 584

Prove that the matrix
$A=\begin{bmatrix} 0 & 1\\ -1& 0 \end{bmatrix}$ is diagonalizable.
Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix.
That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal matrix.

## Problem 583

Consider the $2\times 2$ complex matrix
$A=\begin{bmatrix} a & b-a\\ 0& b \end{bmatrix}.$

(a) Find the eigenvalues of $A$.

(b) For each eigenvalue of $A$, determine the eigenvectors.

(c) Diagonalize the matrix $A$.

(d) Using the result of the diagonalization, compute and simplify $A^k$ for each positive integer $k$.

## Problem 582

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

Suppose that $A$ is a nilpotent matrix and let $B$ be an invertible matrix of the same size as $A$.
Is the matrix $B-A$ invertible? If so prove it. Otherwise, give a counterexample.

## 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 580

(a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly dependent.

(b) Prove that the row vectors of every $5\times 3$ matrix $B$ are linearly dependent.

## Problem 579

Determine whether each of the following sets is a basis for $\R^3$.

(a) $S=\left\{\, \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix} \,\right\}$

(b) $S=\left\{\, \begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix}, \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix}, \begin{bmatrix} 3 \\ 6 \\ 9 \end{bmatrix} \,\right\}$

(c) $S=\left\{\, \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 7 \end{bmatrix} \,\right\}$

(d) $S=\left\{\, \begin{bmatrix} 1 \\ 2 \\ 5 \end{bmatrix}, \begin{bmatrix} 7 \\ 4 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 8 \\ 6 \end{bmatrix}, \begin{bmatrix} -1 \\ 9 \\ 10 \end{bmatrix} \,\right\}$

## 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.

## Problem 574

Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$.

(a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$.

(b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of $\R^3$.

## Problem 572

The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.
There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).
The time limit was 55 minutes.

This post is Part 3 and contains Problem 7, 8, and 9.
Check out Part 1 and Part 2 for the rest of the exam problems.

Problem 7. Let $A=\begin{bmatrix} -3 & -4\\ 8& 9 \end{bmatrix}$ and $\mathbf{v}=\begin{bmatrix} -1 \\ 2 \end{bmatrix}$.

(a) Calculate $A\mathbf{v}$ and find the number $\lambda$ such that $A\mathbf{v}=\lambda \mathbf{v}$.

(b) Without forming $A^3$, calculate the vector $A^3\mathbf{v}$.

Problem 8. Prove that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular.

Problem 9.
Determine whether each of the following sentences is true or false.

(a) There is a $3\times 3$ homogeneous system that has exactly three solutions.

(b) If $A$ and $B$ are $n\times n$ symmetric matrices, then the sum $A+B$ is also symmetric.

(c) If $n$-dimensional vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are linearly dependent, then the vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4$ is also linearly dependent for any $n$-dimensional vector $\mathbf{v}_4$.

(d) If the coefficient matrix of a system of linear equations is singular, then the system is inconsistent.

(e) The vectors
$\mathbf{v}_1=\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$ are linearly independent.

## Problem 571

The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.
There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).
The time limit was 55 minutes.

This post is Part 2 and contains Problem 4, 5, and 6.
Check out Part 1 and Part 3 for the rest of the exam problems.

Problem 4. Let
$\mathbf{a}_1=\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \mathbf{a}_2=\begin{bmatrix} 2 \\ -1 \\ 4 \end{bmatrix}, \mathbf{b}=\begin{bmatrix} 0 \\ a \\ 2 \end{bmatrix}.$

Find all the values for $a$ so that the vector $\mathbf{b}$ is a linear combination of vectors $\mathbf{a}_1$ and $\mathbf{a}_2$.

Problem 5.
Find the inverse matrix of
$A=\begin{bmatrix} 0 & 0 & 2 & 0 \\ 0 &1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{bmatrix}$ if it exists. If you think there is no inverse matrix of $A$, then give a reason.

Problem 6.
Consider the system of linear equations
\begin{align*}
3x_1+2x_2&=1\\
5x_1+3x_2&=2.
\end{align*}

(a) Find the coefficient matrix $A$ of the system.

(b) Find the inverse matrix of the coefficient matrix $A$.

(c) Using the inverse matrix of $A$, find the solution of the system.

(Linear Algebra Midterm Exam 1, the Ohio State University)

## Problem 570

The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.
There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).
The time limit was 55 minutes.

This post is Part 1 and contains the first three problems.
Check out Part 2 and Part 3 for the rest of the exam problems.

Problem 1. Determine all possibilities for the number of solutions of each of the systems of linear equations described below.

(a) A consistent system of $5$ equations in $3$ unknowns and the rank of the system is $1$.

(b) A homogeneous system of $5$ equations in $4$ unknowns and it has a solution $x_1=1$, $x_2=2$, $x_3=3$, $x_4=4$.

Problem 2. Consider the homogeneous system of linear equations whose coefficient matrix is given by the following matrix $A$. Find the vector form for the general solution of the system.
$A=\begin{bmatrix} 1 & 0 & -1 & -2 \\ 2 &1 & -2 & -7 \\ 3 & 0 & -3 & -6 \\ 0 & 1 & 0 & -3 \end{bmatrix}.$

Problem 3. Let $A$ be the following invertible matrix.
$A=\begin{bmatrix} -1 & 2 & 3 & 4 & 5\\ 6 & -7 & 8& 9& 10\\ 11 & 12 & -13 & 14 & 15\\ 16 & 17 & 18& -19 & 20\\ 21 & 22 & 23 & 24 & -25 \end{bmatrix}$ Let $I$ be the $5\times 5$ identity matrix and let $B$ be a $5\times 5$ matrix.
Suppose that $ABA^{-1}=I$.
Then determine the matrix $B$.

(Linear Algebra Midterm Exam 1, the Ohio State University)

## Problem 569

For an $m\times n$ matrix $A$, we denote by $\mathrm{rref}(A)$ the matrix in reduced row echelon form that is row equivalent to $A$.
For example, consider the matrix $A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &2 &2 \end{bmatrix}$
Then we have
$A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &2 &2 \end{bmatrix} \xrightarrow{\frac{1}{2}R_2} \begin{bmatrix} 1 & 1 & 1 \\ 0 &1 & 1 \end{bmatrix} \xrightarrow{R_1-R_2} \begin{bmatrix} 1 & 0 & 0 \\ 0 &1 &1 \end{bmatrix}$ and the last matrix is in reduced row echelon form.
Hence $\mathrm{rref}(A)=\begin{bmatrix} 1 & 0 & 0 \\ 0 &1 &1 \end{bmatrix}$.

Find an example of matrices $A$ and $B$ such that
$\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B).$

## Problem 567

Prove the following identity for any positive integer $n$.
$\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix}^n=\begin{bmatrix} \cos n\theta & -\sin n\theta\\ \sin n\theta& \cos n\theta \end{bmatrix}.$

## Problem 565

Let $I$ be the $2\times 2$ identity matrix.
Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$.

## Problem 564

Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$.

(a) Prove that $A+B$ is skew-symmetric.

(b) Prove that $cA$ is skew-symmetric for any scalar $c$.

(c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is skew-symmetric.

(d) Suppose that $A$ is real skew-symmetric. Prove that $iA$ is an Hermitian matrix.

(e) Prove that if $AB=-BA$, then $AB$ is a skew-symmetric matrix.

(f) Let $\mathbf{v}$ be an $n$-dimensional column vecotor. Prove that $\mathbf{v}^{\trans}A\mathbf{v}=0$.

(g) Suppose that $A$ is a real skew-symmetric matrix and $A^2\mathbf{v}=\mathbf{0}$ for some vector $\mathbf{v}\in \R^n$. Then prove that $A\mathbf{v}=\mathbf{0}$.

## Problem 563

Let
$\mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ a \\ 5 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 4 \\ b \end{bmatrix}$ be vectors in $\R^3$.

Determine a condition on the scalars $a, b$ so that the set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly dependent.

## Problem 562

An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$.
Using the definition of a nonsingular matrix, prove the following statements.

(a) If $A$ and $B$ are $n\times n$ nonsingular matrix, then the product $AB$ is also nonsingular.

(b) Let $A$ and $B$ be $n\times n$ matrices and suppose that the product $AB$ is nonsingular. Then:

1. The matrix $B$ is nonsingular.
2. The matrix $A$ is nonsingular. (You may use the fact that a nonsingular matrix is invertible.)

## Problem 561

Let $A$ be a singular $n\times n$ matrix.
Let
$\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, \dots, \mathbf{e}_n=\begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}$ be unit vectors in $\R^n$.

Prove that at least one of the following matrix equations
$A\mathbf{x}=\mathbf{e}_i$ for $i=1,2,\dots, n$, must have no solution $\mathbf{x}\in \R^n$.