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 $A+B\in W$.