<s>[INST] Is a symmetric matrix a subspace of nxn matrices? Let $M$ be a vector space for all 2x2 matrices.
Show that the set of all the symmetric matrices $M2$={$B ∈ M2 : B=B^t$} is a subspace of M.

My solution:

*

*The null matrix is symmetric

$$B=\begin{pmatrix}0&0\\0&0\end{pmatrix}$$



*Let $B$, $C$ $∈$ $M2$, where $B = B^t$, $C=C^t$. Then, $B+C=B^t + C^t$.

$$B = \begin{pmatrix}a&c\\c&b\end{pmatrix}$$
$$C = \begin{pmatrix}d&f\\f&e\end{pmatrix}$$
$$B+C = \begin{pmatrix}a+d&c+f\\c+f&b+e\end{pmatrix}$$
$$B^t = \begin{pmatrix}a&c\\c&b\end{pmatrix}$$
$$C^t = \begin{pmatrix}d&f\\f&e\end{pmatrix}$$
$$B^t+C^t = \begin{pmatrix}a+d&c+f\\c+f&b+e\end{pmatrix}$$
Therefore, $B + C ∈ M2$.



*Let $B ∈ M2$, $θ ∈ R$. Then, $θB = θB^t$
$$B = \begin{pmatrix}a&c\\c&b\end{pmatrix}$$ $$θB = \begin{pmatrix}θa&θc\\θc&θb\end{pmatrix}$$
$$B^t = \begin{pmatrix}a&c\\c&b\end{pmatrix}$$ $$θB^t = \begin{pmatrix}θa&θc\\θc&θb\end{pmatrix}$$
Therefore, $θB ∈ M2$.

So, $M2$ is, indeed, a subspace of $M$.
I am just learning linear algebra, so I apologize if I've made any mistakes.
Also, I just got into this community, so I am still learning the formatting, so I apologize for any mistakes as well.
Appreciate all the answers.
 [/INST] A simpler way to prove this is to note that if $A,B$ are symmetric matrices then since $(\lambda A)^T = \lambda A^T = \lambda A$ and $(A+B)^T = A^T+B^T = A +B$, so are $\lambda A$ and $A+B$.
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