## If Matrices Commute $AB=BA$, then They Share a Common Eigenvector

## Problem 608

Let $A$ and $B$ be $n\times n$ matrices and assume that they commute: $AB=BA$.

Then prove that the matrices $A$ and $B$ share at least one common eigenvector.

Let $A$ and $B$ be $n\times n$ matrices and assume that they commute: $AB=BA$.

Then prove that the matrices $A$ and $B$ share at least one common eigenvector.

Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less.

Let

\[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\]
where

\begin{align*}

p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\

p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3.

\end{align*}

**(a)** Find a basis $Q$ of the span $\Span(S)$ consisting of polynomials in $S$.

**(b)** For each polynomial in $S$ that is not in $Q$, find the coordinate vector with respect to the basis $Q$.

*(The Ohio State University, Linear Algebra Midterm)*

Read solution

Let $V$ be a vector space and $B$ be a basis for $V$.

Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$.

Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ with respect to the basis $B$.

After applying the elementary row operations to $A$, we obtain the following matrix in reduced row echelon form

\[\begin{bmatrix}

1 & 0 & 2 & 1 & 0 \\

0 & 1 & 3 & 0 & 1 \\

0 & 0 & 0 & 0 & 0 \\

0 & 0 & 0 & 0 & 0

\end{bmatrix}.\]

**(a)** What is the dimension of $V$?

**(b)** What is the dimension of $\Span\{\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5\}$?

*(The Ohio State University, Linear Algebra Midterm)*

Read solution

Let $T:\R^2 \to \R^3$ be a linear transformation such that

\[T\left(\, \begin{bmatrix}

3 \\

2

\end{bmatrix} \,\right)

=\begin{bmatrix}

1 \\

2 \\

3

\end{bmatrix} \text{ and }

T\left(\, \begin{bmatrix}

4\\

3

\end{bmatrix} \,\right)

=\begin{bmatrix}

0 \\

-5 \\

1

\end{bmatrix}.\]

**(a)** Find the matrix representation of $T$ (with respect to the standard basis for $\R^2$).

**(b)** Determine the rank and nullity of $T$.

*(The Ohio State University, Linear Algebra Midterm)*

Read solution

Let

\[A=\begin{bmatrix}

1 & -1 & 0 & 0 \\

0 &1 & 1 & 1 \\

1 & -1 & 0 & 0 \\

0 & 2 & 2 & 2\\

0 & 0 & 0 & 0

\end{bmatrix}.\]

**(a)** Find a basis for the null space $\calN(A)$.

**(b)** Find a basis of the range $\calR(A)$.

**(c)** Find a basis of the row space for $A$.

*(The Ohio State University, Linear Algebra Midterm)*

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Let $C[-2\pi, 2\pi]$ be the vector space of all continuous functions defined on the interval $[-2\pi, 2\pi]$.

Consider the functions \[f(x)=\sin^2(x) \text{ and } g(x)=\cos^2(x)\]
in $C[-2\pi, 2\pi]$.

Prove or disprove that the functions $f(x)$ and $g(x)$ are linearly independent.

*(The Ohio State University, Linear Algebra Midterm)*

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Let $W$ be a subspace of $\R^4$ with a basis

\[\left\{\, \begin{bmatrix}

1 \\

0 \\

1 \\

1

\end{bmatrix}, \begin{bmatrix}

0 \\

1 \\

1 \\

1

\end{bmatrix} \,\right\}.\]

Find an ortho**normal** basis of $W$.

*(The Ohio State University, Linear Algebra Midterm)*

Read solution

Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers.

Let

\[W=\left\{\, A\in V \quad \middle | \quad A=\begin{bmatrix}

a & b\\

c& -a

\end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\]

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

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

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

*(The Ohio State University, Linear Algebra Midterm)*

Read solution

Let $\mathbf{v}_1=\begin{bmatrix}

2/3 \\ 2/3 \\ 1/3

\end{bmatrix}$ be a vector in $\R^3$.

Find an orthonormal basis for $\R^3$ containing the vector $\mathbf{v}_1$.

Add to solve laterLet $A$ be a real symmetric matrix whose diagonal entries are all positive real numbers.

Is it true that the all of the diagonal entries of the inverse matrix $A^{-1}$ are also positive?

If so, prove it. Otherwise, give a counterexample.

Let $R$ be a commutative ring with $1$.

Prove that if every proper ideal of $R$ is a prime ideal, then $R$ is a field.

Add to solve laterLet $F:\R^2\to \R^2$ be the function that maps each vector in $\R^2$ to its reflection with respect to $x$-axis.

Determine the formula for the function $F$ and prove that $F$ is a linear transformation.

Add to solve later Let

\[A=\begin{bmatrix}

a & b\\

-b& a

\end{bmatrix}\]
be a $2\times 2$ matrix, where $a, b$ are real numbers.

Suppose that $b\neq 0$.

Prove that the matrix $A$ does not have real eigenvalues.

Add to solve laterLet $U$ and $V$ be subspaces of the $n$-dimensional vector space $\R^n$.

Prove that the intersection $U\cap V$ is also a subspace of $\R^n$.

Add to solve later Is it possible that each element of an infinite group has a finite order?

If so, give an example. Otherwise, prove the non-existence of such a group.

We fix a nonzero vector $\mathbf{a}$ in $\R^3$ and define a map $T:\R^3\to \R^3$ by

\[T(\mathbf{v})=\mathbf{a}\times \mathbf{v}\]
for all $\mathbf{v}\in \R^3$.

Here the right-hand side is the cross product of $\mathbf{a}$ and $\mathbf{v}$.

**(a)** Prove that $T:\R^3\to \R^3$ is a linear transformation.

**(b)** Determine the eigenvalues and eigenvectors of $T$.

Let $\R^n$ be an inner product space with inner product $\langle \mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^{\trans}\mathbf{y}$ for $\mathbf{x}, \mathbf{y}\in \R^n$.

A linear transformation $T:\R^n \to \R^n$ is called **orthogonal transformation** if for all $\mathbf{x}, \mathbf{y}\in \R^n$, it satisfies

\[\langle T(\mathbf{x}), T(\mathbf{y})\rangle=\langle\mathbf{x}, \mathbf{y} \rangle.\]

Prove that if $T:\R^n\to \R^n$ is an orthogonal transformation, then $T$ is an isomorphism.

Add to solve later Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ be a set of nonzero vectors in $\R^n$.

Suppose that $S$ is an orthogonal set.

**(a)** Show that $S$ is linearly independent.

**(b)** If $k=n$, then prove that $S$ is a basis for $\R^n$.

Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let

\[V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}\]
be a subset in $C[-1, 1]$.

**(a)** Prove that $V$ is a subspace of $C[-1, 1]$.

**(b)** Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$.

**(c)** Prove that

\[B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}\]
is a basis for $V$.

Let $R$ be an integral domain and let $I$ be an ideal of $R$.

Is the quotient ring $R/I$ an integral domain?