We verify the subspace criteria: the zero vector of $V$ is in $W$, and $W$ is closed under addition and scalar multiplication.

First, let $\mathbf{0} \in V$ be the $k\times k$ zero matrix. Then $ R \mathbf{0} S = \mathbf{0}$, and so $\mathbf{0} \in W$.

Now suppose $X, Y \in W$. Then there are elements $A, B \in V$ such that $RAS = X$ and $RBS = Y$. Then
\[X + Y = RAS + RBS = R (A+B) S\]
and so $X+Y \in W$.

Now for a scalar $c \in \mathbb{R}$ and matrix $X = RAS \in W$, we have
\[cX = c RAS = R (cA )S,\]
and so $cX \in W$ as well.

Prove that the Center of Matrices is a Subspace
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 […]

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 […]

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 […]

The Intersection of Two Subspaces is also a Subspace
Let $U$ and $V$ be subspaces of the $n$-dimensional vector space $\R^n$.
Prove that the intersection $U\cap V$ is also a subspace of $\R^n$.
Definition (Intersection).
Recall that the intersection $U\cap V$ is the set of elements that are both elements of $U$ […]

The Subspace of Linear Combinations whose Sums of Coefficients are zero
Let $V$ be a vector space over a scalar field $K$.
Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset
\[W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } […]

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 & […]

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 […]