## 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.

of the day

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 $T:\R^2 \to \R^3$ 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)*

Read solution

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 $W$.

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 $V$.

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 laterLet $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ 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 $\R^3$.

Let $T:\R^3 \to \R^3$ be the linear transformation defined by the formula

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

x_1 \\

x_2 \\

x_3

\end{bmatrix} \,\right)=\begin{bmatrix}

x_1+3x_2-2x_3 \\

2x_1+3x_2 \\

x_2+x_3

\end{bmatrix}.\]

Determine whether $T$ is an isomorphism and if so find the formula for the inverse linear transformation $T^{-1}$.

Add to solve later Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$.

Consider a linear transformation $T:U\to V$.

Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one).

Add to solve later Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less.

Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where

\begin{align*}

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

p_3(x)&=1+2x+8x^2, \quad p_4(x)=1+x+x^2.

\end{align*}

**(a)** Find a basis of $P_2$ among the vectors of $S$. (Explain why it is a basis of $P_2$.)

**(b)** Let $B’$ be the basis you obtained in part (a).

For each vector of $S$ which is not in $B’$, find the coordinate vector of it with respect to the basis $B’$.

(*The Ohio State University, Linear Algebra Final Exam Problem*)

Let $T:\R^2 \to \R^3$ be a linear transformation given by

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

x_1 \\

x_2

\end{bmatrix} \,\right)

=

\begin{bmatrix}

x_1-x_2 \\

x_2 \\

x_1+ x_2

\end{bmatrix}.\]
Find an orthonormal basis of the range of $T$.

(*The Ohio State University, Linear Algebra Final Exam Problem*)

Let

\[A=\begin{bmatrix}

1 & 2 & 1 \\

-1 &4 &1 \\

2 & -4 & 0

\end{bmatrix}.\]
The matrix $A$ has an eigenvalue $2$.

Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$.

(*The Ohio State University, Linear Algebra Final Exam Problem*)

Determine all linear transformations of the $2$-dimensional $x$-$y$ plane $\R^2$ that take the line $y=x$ to the line $y=-x$.

Add to solve later Let $\mathbf{u}=\begin{bmatrix}

1 \\

1 \\

0

\end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation

\[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\]

**(a)** Calculate the null space $\calN(T)$, a basis for $\calN(T)$ and nullity of $T$.

**(b)** Only by using part (a) and no other calculations, find $\det(A)$, where $A$ is the matrix representation of $T$ with respect to the standard basis of $\R^3$.

**(c)** Calculate the range $\calR(T)$, a basis for $\calR(T)$ and the rank of $T$.

**(d)** Calculate the matrix $A$ representing $T$ with respect to the standard basis for $\R^3$.

**(e)** Let

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

1 \\

0 \\

0

\end{bmatrix}, \begin{bmatrix}

-1 \\

1 \\

0

\end{bmatrix}, \begin{bmatrix}

0 \\

-1 \\

1

\end{bmatrix} \,\right\}\]
be a basis for $\R^3$.

Calculate the coordinates of $\begin{bmatrix}

x \\

y \\

z

\end{bmatrix}$ with respect to $B$.

(*The Ohio State University, Linear Algebra Exam Problem*)

Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$.

Then prove that

\[\dim(U+V) \leq \dim(U)+\dim(V).\]

Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.

Define the map $f:\R^2 \to \calF[0, 2\pi]$ by

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

\alpha \\

\beta

\end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta \sin x.\]
We put

\[V:=\im f=\{\alpha \cos x + \beta \sin x \in \calF[0, 2\pi] \mid \alpha, \beta \in \R\}.\]

**(a)** Prove that the map $f$ is a linear transformation.

**(b)** Prove that the set $\{\cos x, \sin x\}$ is a basis of the vector space $V$.

**(c)** Prove that the kernel is trivial, that is, $\ker f=\{\mathbf{0}\}$.

(This yields an isomorphism of $\R^2$ and $V$.)

**(d)** Define a map $g:V \to V$ by

\[g(\alpha \cos x + \beta \sin x):=\frac{d}{dx}(\alpha \cos x+ \beta \sin x)=\beta \cos x -\alpha \sin x.\]
Prove that the map $g$ is a linear transformation.

**(e)** Find the matrix representation of the linear transformation $g$ with respect to the basis $\{\cos x, \sin x\}$.

(Kyoto University, Linear Algebra exam problem)

Add to solve later Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$.

Let $A \in V$ and consider the set

\[S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}\]
of $n^2$ elements.

Prove that the set $S_A$ cannot be a basis of the vector space $V$ for any $A\in V$.

Let $A=\begin{bmatrix}

1 & 0 & 1 \\

0 &1 &0

\end{bmatrix}$.

**(a)** Find an orthonormal basis of the null space of $A$.

**(b)** Find the rank of $A$.

**(c)** Find an orthonormal basis of the row space of $A$.

(*The Ohio State University, Linear Algebra Exam Problem*)

Read solution

**(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).\]