# Tagged: counterexample

## Problem 637

Let $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors.

(a) Prove that $\mathbf{v}^\trans \mathbf{w} = \mathbf{w}^\trans \mathbf{v}$.

(b) Provide an example to show that $\mathbf{v} \mathbf{w}^\trans$ is not always equal to $\mathbf{w} \mathbf{v}^\trans$.

## Problem 634

Let $A$ and $B$ be $n \times n$ matrices.

Is it always true that $\tr (A B) = \tr (A) \tr (B)$?

If it is true, prove it. If not, give a counterexample.

## 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 439

Is every diagonalizable matrix invertible?

## Problem 354

Let $G$ be a group. Let $a$ and $b$ be elements of $G$.
If the order of $a, b$ are $m, n$ respectively, then is it true that the order of the product $ab$ divides $mn$? If so give a proof. If not, give a counterexample.

## Problem 338

Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.
(1) $S_1=\left \{\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}$ in the vector space $\R^3$.

(2) $S_2=\left \{\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 \quad \middle | \quad x_1-4x_2+5x_3=2 \,\right \}$ in the vector space $\R^3$.

(3) $S_3=\left \{\, \begin{bmatrix} x \\ y \end{bmatrix}\in \R^2 \quad \middle | \quad y=x^2 \quad \,\right \}$ in the vector space $\R^2$.

(4) Let $P_4$ be the vector space of all polynomials of degree $4$ or less with real coefficients.
$S_4=\{ f(x)\in P_4 \mid f(1) \text{ is an integer}\}$ in the vector space $P_4$.

(5) $S_5=\{ f(x)\in P_4 \mid f(1) \text{ is a rational number}\}$ in the vector space $P_4$.

(6) Let $M_{2 \times 2}$ be the vector space of all $2\times 2$ real matrices.
$S_6=\{ A\in M_{2\times 2} \mid \det(A) \neq 0\}$ in the vector space $M_{2\times 2}$.

(7) $S_7=\{ A\in M_{2\times 2} \mid \det(A)=0\}$ in the vector space $M_{2\times 2}$.

(Linear Algebra Exam Problem, the Ohio State University)

(8) Let $C[-1, 1]$ be the vector space of all real continuous functions defined on the interval $[a, b]$.
$S_8=\{ f(x)\in C[-2,2] \mid f(-1)f(1)=0\}$ in the vector space $C[-2, 2]$.

(9) $S_9=\{ f(x) \in C[-1, 1] \mid f(x)\geq 0 \text{ for all } -1\leq x \leq 1\}$ in the vector space $C[-1, 1]$.

(10) Let $C^2[a, b]$ be the vector space of all real-valued functions $f(x)$ defined on $[a, b]$, where $f(x), f'(x)$, and $f^{\prime\prime}(x)$ are continuous on $[a, b]$. Here $f'(x), f^{\prime\prime}(x)$ are the first and second derivative of $f(x)$.
$S_{10}=\{ f(x) \in C^2[-1, 1] \mid f^{\prime\prime}(x)+f(x)=\sin(x) \text{ for all } -1\leq x \leq 1\}$ in the vector space $C[-1, 1]$.

(11) Let $S_{11}$ be the set of real polynomials of degree exactly $k$, where $k \geq 1$ is an integer, in the vector space $P_k$.

(12) Let $V$ be a vector space and $W \subset V$ a vector subspace. Define the subset $S_{12}$ to be the complement of $W$,
$V \setminus W = \{ \mathbf{v} \in V \mid \mathbf{v} \not\in W \}.$

## Problem 253

Determine whether the following is true or false. If it is true, then give a proof. If it is false, then give a counterexample.

Let $W_1$ and $W_2$ be subspaces of the vector space $\R^n$.
If $B_1$ and $B_2$ are bases for $W_1$ and $W_2$, respectively, then $B_1\cap B_2$ is a basis of the subspace $W_1\cap W_2$.

## Problem 186

Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an integer greater than $1$.

Is it true that
$\det(A+B)=\det(A)+\det(B)?$ If so, then give a proof. If not, then give a counterexample.

## Problem 98

Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix.

Is it true that the matrix product with opposite order $BA$ is also the zero matrix?
If so, give a proof. If not, give a counterexample.

## 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.