I am trying to prove that W is a subspace of V with:

$V = M_{n\times n}$,

$W = \{A \in M_{n\times n} : A = -A^T\}$

I am fairly sure $W$ is closed under addition, but am not sure how to prove it for all $M_{n\times n}$

I can prove it for $M_{2\times 2}$:

Let $A = \begin{bmatrix}0 & a\\-a & 0\end{bmatrix} \in W$

Let $B = \begin{bmatrix}0 & b\\-b & 0\end{bmatrix} \in W$

$A + B = \begin{bmatrix}0 & a\\-a & 0\end{bmatrix} + \begin{bmatrix}0 & b\\-b & 0\end{bmatrix}$

$ = \begin{bmatrix}0 & a + b\\-a-b & 0\end{bmatrix} = \begin{bmatrix}0 & a + b\\-(a+b) & 0\end{bmatrix} \in W$

How could I generalise this to all $A, B \in M_{n\times n}$

## Best Answer

Let $A,B\in W$. Then $$-(A+B)^T=-(A^T+B^T)=-A^T-B^T=A+B$$ The final equality comes from $A=-A^T$ and $B=-B^T$ (because they are in $W$). So $A+B=-(A+B)^T$ so $A+B\in W$.