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 $V$.

We must show that $W$ satisfies the three criteria for vector subspaces.
Namely, the zero vector of $V$ is in $W$, and $W$ is closed under addition and scalar multiplication.

First, the zero element in $V$ is the matrix $\mathbf{0}$ whose entries are all $0$. For any other matrix $\mathbf{x} \in V$, we have $\mathbf{0} \mathbf{x} = \mathbf{0} = \mathbf{x} \mathbf{0}$. So we see that $\mathbf{0} \in W$.

Now suppose $\mathbf{v}, \mathbf{w} \in W$ and $c \in \mathbb{R}$. Then for any $\mathbf{x} \in V$, we have
\[(\mathbf{v} + \mathbf{w} ) \mathbf{x} = \mathbf{v} \mathbf{x} + \mathbf{w} \mathbf{x} = \mathbf{x} \mathbf{v} + \mathbf{x} \mathbf{w} = \mathbf{x} ( \mathbf{v} + \mathbf{w} ),\]
where the second equality follows because $\mathbf{v}$ and $\mathbf{w}$ lie in $W$. So we see that $\mathbf{v} + \mathbf{w} \in W$ as well, and so $W$ is closed under addition.

Finally we must show that $c \mathbf{v} \in W$ as well. For any other $\mathbf{x} \in V$, we have
\[(c \mathbf{v} ) \mathbf{x} = c ( \mathbf{v} \mathbf{x}) = c ( \mathbf{x} \mathbf{v} ) = \mathbf{x} ( c \mathbf{v} ),\]
where the second equality follows from the fact that $\mathbf{v} \in W$ and so $\mathbf{v} \mathbf{x} = \mathbf{x} \mathbf{v}$.

Thus we see that $c \mathbf{v} \in W$, finishing the proof.

For Fixed Matrices $R, S$, the Matrices $RAS$ form a Subspace
Let $V$ be the vector space of $k \times k$ matrices. Then for fixed matrices $R, S \in V$, define the subset $W = \{ R A S \mid A \in V \}$.
Prove that $W$ is a vector subspace of $V$.
Proof.
We verify the subspace criteria: the zero vector of $V$ is in $W$, and […]

The Centralizer of a Matrix is a Subspace
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 $V$.
Proof.
First we check that the zero […]

Subspaces of Symmetric, Skew-Symmetric Matrices
Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.
(a) The set $S$ consisting of all $n\times n$ symmetric matrices.
(b) The set $T$ consisting of […]

Determine Whether a Set of Functions $f(x)$ such that $f(x)=f(1-x)$ is a Subspace
Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let
\[W=\{ f(x)\in V \mid f(x)=f(1-x) \text{ for } x\in [0,1]\}\]
be a subset of $V$. Determine whether the subset $W$ is a subspace of the vector space $V$.
Proof. […]

Determine the Values of $a$ so that $W_a$ is a Subspace
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?
Solution.
The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by […]

Sequences Satisfying Linear Recurrence Relation Form a Subspace
Let $V$ be a real vector space of all real sequences
\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \cdots).\]
Let $U$ be the subset of $V$ defined by
\[U=\{ (a_i)_{i=1}^{\infty} \in V \mid a_{k+2}-5a_{k+1}+3a_{k}=0, k=1, 2, \dots \}.\]
Prove that $U$ is a subspace of […]

The Vector Space Consisting of All Traceless Diagonal Matrices
Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero.
That is,
\begin{equation*}
V:=\left\{ A=\begin{bmatrix}
a_{11} & 0 & \dots & 0 \\
0 &a_{22} & \dots & 0 \\
0 & 0 & \ddots & \vdots \\
0 & 0 & \dots & […]