Dimension of tracefree matrix subspace. Let $P\in M_n(\Bbb R)$ be an invertible matrix.
Find the dimension of the following subspace:
$$L = \{ X \in M_n(\Bbb R)|  tr(PX)=0\}$$
Don't know where to start. Any help?

The primary observation in this problem is that $tr(PX) = \sum_{1\le i, j \le n} P_{ij}X_{ij}$, which is same as the dot product in the case of vectors. The dimension of the vector space $M_n(\mathbb{R})$ is $n^2$. We need to find the dimension of the subspace which is perpendicular to $P$ (since the $dot$ product is zero). By drawing an analogy with vectors, its easy to see that the dimension of the subspace is $n^2-1$.
