## Determine the Values of $a$ so that $W_a$ is a Subspace

## Problem 662

For what real values of $a$ is the set

\[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\]
a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?

For what real values of $a$ is the set

\[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\]
a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?

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 laterLet $V$ 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 laterFix 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 $\R^3$.

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 laterSuppose 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 laterSuppose 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 laterConsider 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.

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 laterWrite the vector $\begin{bmatrix} 1 \\ 3 \\ -1 \end{bmatrix}$ as a linear combination of the vectors

\[\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \, \begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix} , \, \begin{bmatrix} 2 \\ 0 \\ 4 \end{bmatrix}.\]

Prove that any set of vectors which contains the zero vector is linearly dependent.

Add to solve later**(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 $g(\pi) = -5$.

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 laterA 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 laterDetermine 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)** $\left[\begin{array}{rr|r} 1 & 2 & 0 \\ 1 & 1 & -1 \end{array} \right]$.

Read solution

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**(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.

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 $A$.

Read solution

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 laterFor 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)** $E = \begin{bmatrix} -2 & 3 & 1 \end{bmatrix}$.