# Category: Linear Algebra

## Problem 661

Let $C(\mathbb{R})$ be the vector space of real-valued functions on $\mathbb{R}$.

Consider the set of functions $W = \{ f(x) = a + b \cos(x) + c \cos(2x) \mid a, b, c \in \mathbb{R} \}$.

Prove that $W$ is a vector subspace of $C(\mathbb{R})$. Add to solve later

## Problem 660

Let $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define
$W = \{ A \in V \mid AM = MA \}.$ The set $W$ here is called the centralizer of $M$ in $V$.

Prove that $W$ is a subspace of $V$. Add to solve later

## Problem 659

Fix the row vector $\mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}$, and let $\R^3$ be the vector space of $3 \times 1$ column vectors. Define
$W = \{ \mathbf{v} \in \R^3 \mid \mathbf{b} \mathbf{v} = 0 \}.$ Prove that $W$ is a vector subspace of $\R^3$. Add to solve later

## Problem 658

Let $V$ be the vector space of $n \times n$ matrices with real coefficients, and define
$W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.$ The set $W$ is called the center of $V$.

Prove that $W$ is a subspace of $V$. Add to solve later

## Problem 657

Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$. Add to solve later

## Problem 656

Suppose that an $n \times m$ matrix $M$ is composed of the column vectors $\mathbf{b}_1 , \cdots , \mathbf{b}_m$.

Prove that a vector $\mathbf{v} \in \R^n$ can be written as a linear combination of the column vectors if and only if there is a vector $\mathbf{x}$ which solves the equation $M \mathbf{x} = \mathbf{v}$. Add to solve later

## Problem 655

Consider the matrix $M = \begin{bmatrix} 1 & 4 \\ 3 & 12 \end{bmatrix}$.

(a) Show that $M$ is singular.

(b) Find a non-zero vector $\mathbf{v}$ such that $M \mathbf{v} = \mathbf{0}$, where $\mathbf{0}$ is the $2$-dimensional zero vector. Add to solve later

## Problem 654

Suppose $M$ is an $n \times n$ upper-triangular matrix.

If the diagonal entries of $M$ are all non-zero, then prove that the column vectors are linearly independent.

Does the conclusion hold if we do not assume that $M$ has non-zero diagonal entries? Add to solve later

## Problem 653

Write the vector $\begin{bmatrix} 1 \\ 3 \\ -1 \end{bmatrix}$ as a linear combination of the vectors
$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \, \begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix} , \, \begin{bmatrix} 2 \\ 0 \\ 4 \end{bmatrix}.$ Add to solve later

## Problem 652

Prove that any set of vectors which contains the zero vector is linearly dependent. Add to solve later

## Problem 651

(a) Find a function
$g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)$ such that $g(0) = g(\pi/2) = g(\pi) = 0$, where $a, b, c$ are constants.

(b) Find real numbers $a, b, c$ such that the function
$g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)$ satisfies $g(0) = 3$, $g(\pi/2) = 1$, and $g(\pi) = -5$. Add to solve later

## Problem 650

Find a quadratic function $f(x) = ax^2 + bx + c$ such that $f(1) = 3$, $f'(1) = 3$, and $f^{\prime\prime}(1) = 2$.

Here, $f'(x)$ and $f^{\prime\prime}(x)$ denote the first and second derivatives, respectively. Add to solve later

## Problem 649

A 2-digit number has two properties: The digits sum to 11, and if the number is written with digits reversed, and subtracted from the original number, the result is 45.

Find the number. Add to solve later

## Problem 648

Determine whether the following augmented matrices are in reduced row echelon form, and calculate the solution sets of their associated systems of linear equations.

(a) $\left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 6 \end{array} \right]$.

(b) $\left[\begin{array}{rrr|r} 1 & 0 & 3 & -4 \\ 0 & 1 & 2 & 0 \end{array} \right]$.

(c) $\left[\begin{array}{rr|r} 1 & 2 & 0 \\ 1 & 1 & -1 \end{array} \right]$. Add to solve later

## Problem 647

Recall that a matrix $A$ is symmetric if $A^\trans = A$, where $A^\trans$ is the transpose of $A$.

Is it true that if $A$ is a symmetric matrix and in reduced row echelon form, then $A$ is diagonal? If so, prove it.

Otherwise, provide a counterexample. Add to solve later

## Problem 646

(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.

(b) Find all such matrices with rank 2. Add to solve later

## Problem 645

Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix.

Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$. Add to solve later

## Problem 644

If $A, B$ have the same rank, can we conclude that they are row-equivalent?

If so, then prove it. If not, then provide a counterexample. Add to solve later

## Problem 643

For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix.

(a) $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$.

(b) $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$.

(c) $C = \begin{bmatrix} 2 & -2 & 4 \\ 4 & 1 & -2 \\ 6 & -1 & 2 \end{bmatrix}$.

(d) $D = \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix}$.

(e) $E = \begin{bmatrix} -2 & 3 & 1 \end{bmatrix}$. Add to solve later

## Problem 642

If $A, B, C$ are three $m \times n$ matrices such that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$, then can we conclude that $A$ is row-equivalent to $C$?

If so, then prove it. If not, then provide a counterexample. Add to solve later