$\mathbf{Question}$: Let $T \in M_{m \times n} (\mathbb{R})$. Let $V$ be the subspace of $M_{n \times p}(\mathbb{R})$ defined by $V=\{X \in M_{n \times p}(\mathbb{R}): TX= 0\}$. Find the dimension of $V$.
$\mathbf{Attempt}:$ We know that the dimension of the solution space of $m$ homogeneous equation in $n$ unknowns is
$rank \ A +dim \ \chi(A)=n$ where $A$ is the coefficient matrix and $\chi(A)$ is the solution space.
Now, each of the columns of $X$, say $C_i$ can be treated as a solution to the homogeneous system $TC_i=0$. Dimension of each solution space is ($n-rank \ T$). There are $p$ such columns, so the answer must be
$p(n-rank \ T)$.
Is this correct? Kindly verify.