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 $V$.

First we check that the zero element of $V$ lies in $W$. The zero element of $V$ is the $n \times n$ zero matrix $\mathbf{0}$.

It is clear that $M \mathbf{0} = \mathbf{0} = \mathbf{0} M$, and so $\mathbf{0} \in W$.

Next suppose $A, B \in W$ and $c \in \mathbb{R}$. Then $AM = MA$ and $BM = MB$, and so
\[( A + B ) M = A M + B M = M A + M B = M ( A + B ).\]
Thus, $A + B \in W$.

We also have
\[( c A ) M = c ( A M ) = c ( M A ) = M ( c A ),\]
and so $c A \in W$.

These three criteria show that $W$ is a subspace of $V$.

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 $U$ […]

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 […]

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 […]

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 […]

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 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 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. […]