$\mathfrak{sp}_4$ is a subspace of the vector space of all $4\times 4$ matrices

Question

Let $\mathfrak{sp}_4$ denote the set of all matrices $X$ satisfying $$X^TM+MX=0$$

How can I show that $\mathfrak{sp}_4$ is a vector subspace of the vector space of all $4\times 4$ matrices?

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I think the only property that I need to worry about satisfying is closure, is this correct?

I mean that should follow from two $4\times 4$ matrices being multiplied together yielding another $4\times 4$ matrix. Perhaps there is an eigenvalue argument?

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$M=\begin{pmatrix}0&0&1&0\\0&0&0&1\\-1&0&0&0\\0&-1&0&0\end{pmatrix}$

Answer 1

To check that a subset $S$ of a vector space (over, say, the field $\mathbb{F}$) is a vector subspace, we need only check:

1. That the subset is nonempty ($S \neq \varnothing$).
2. That the subset is closed under scalar multiplication (for all $f \in \mathbb{F}$, $s \in S$, we have $fs \in S$).
3. That the subset is closed under addition (for all $s_1, s_2 \in S$, we have $s_1 + s_2 \in S$).