## Use Coordinate Vectors to Show a Set is a Basis for the Vector Space of Polynomials of Degree 2 or Less

## Problem 588

Let $P_2$ be the vector space over $\R$ of all polynomials of degree $2$ or less.

Let $S=\{p_1(x), p_2(x), p_3(x)\}$, where

\[p_1(x)=x^2+1, \quad p_2(x)=6x^2+x+2, \quad p_3(x)=3x^2+x.\]

**(a)** Use the basis $B=\{x^2, x, 1\}$ of $P_2$ to prove that the set $S$ is a basis for $P_2$.

**(b)** Find the coordinate vector of $p(x)=x^2+2x+3\in P_2$ with respect to the basis $S$.

## Commuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same Eigenvalues

## Problem 587

Let $A$ and $B$ be square matrices such that they commute each other: $AB=BA$.

Assume that $A-B$ is a nilpotent matrix.

Then prove that the eigenvalues of $A$ and $B$ are the same.

Add to solve later## The set of $2\times 2$ Symmetric Matrices is a Subspace

## Problem 586

Let $V$ be the vector space over $\R$ of all real $2\times 2$ matrices.

Let $W$ be the subset of $V$ consisting of all symmetric matrices.

**(a)** Prove that $W$ is a subspace of $V$.

**(b)** Find a basis of $W$.

**(c)** Determine the dimension of $W$.

## Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix

## Problem 585

Consider the Hermitian matrix

\[A=\begin{bmatrix}

1 & i\\

-i& 1

\end{bmatrix}.\]

**(a)** Find the eigenvalues of $A$.

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

**(c)** Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix $D$ and a unitary matrix $U$ such that $U^{-1}AU=D$.

## A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix

## 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.

## Diagonalize the Upper Triangular Matrix and Find the Power of the 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 $k$.

## Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible?

## 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.

## The Subspace of Linear Combinations whose Sums of Coefficients are zero

## 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 $V$.

Add to solve later## The Column Vectors of Every $3\times 5$ Matrix Are Linearly Dependent

## 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.

## Determine Whether Each Set is a Basis for $\R^3$

## 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\}$

## Find the Dimension of the Subspace of Vectors Perpendicular to Given Vectors

## 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 $V$.

## Every Basis of a Subspace Has the Same Number of Vectors

## 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 $V$.

Add to solve later## If there are More Vectors Than a Spanning Set, then Vectors are Linearly Dependent

## 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.

Add to solve later## If a Half of a Group are Elements of Order 2, then the Rest form an Abelian Normal Subgroup of Odd Order

## Problem 575

Let $G$ be a finite group of order $2n$.

Suppose that exactly a half of $G$ consists of elements of order $2$ and the rest forms a subgroup.

Namely, suppose that $G=S\sqcup H$, where $S$ is the set of all elements of order in $G$, and $H$ is a subgroup of $G$. The cardinalities of $S$ and $H$ are both $n$.

Then prove that $H$ is an abelian normal subgroup of odd order.

Add to solve later## Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis.

## 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 $\R^3$.

## Polynomial Ring with Integer Coefficients and the Prime Ideal $I=\{f(x) \in \Z[x] \mid f(-2)=0\}$

## Problem 573

Let $\Z[x]$ be the ring of polynomials with integer coefficients.

Prove that

\[I=\{f(x)\in \Z[x] \mid f(-2)=0\}\]
is a prime ideal of $\Z[x]$. Is $I$ a maximal ideal of $\Z[x]$?

## Linear Algebra Midterm 1 at the Ohio State University (3/3)

## 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.

## Linear Algebra Midterm 1 at the Ohio State University (2/3)

## 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*)

Read solution

## Linear Algebra Midterm 1 at the Ohio State University (1/3)

## 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*)

Read solution

## An Example of Matrices $A$, $B$ such that $\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B)$

## 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

\[\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B).\]