Problem: Let $X$ be a matrix where $x_{ij}$ are all variables. $A$ is invertible and both matrices are real and $n \times n$. What is the dimension of the subspace of X satisfying the equation: $A^TX + X^TA = 0$
What I did: $A^TX = M$, then
$M^t = X^TA$ so $M = -M^T$ There, I found out that $m_{ij} = -m_{ji}$ and $m_{ii} = 0$
Therefore, if $X_{i}$ are the columns of $X$ and $A_{i}$ the columns of $A$;
$(A_{i}\cdot X_{j}) = m_{ij}$ and then:
$(A_{i}\cdot X_{j}) + (A_{j}\cdot X_{i}) = 0$
Giving me $(n^2 + n)/2$ and $n^2$ variables
Question: How can I find the dimension? My guess is that every equation is independent, so the dimension would be $(n^2-n)/2$, but I was not able to prove this fact so far.