Q.2 : Let $T \in \textsf{M}_{m \times n} (\mathbb R)$. Let $\textsf V$ be a subspace of $\textsf{M}_{n \times p} (\mathbb R)$ defined by $$\textsf V = \{ X \in \textsf{M}_{m \times n} (\mathbb R) : \, TX = 0 \}.$$ Then the dimension of $\textsf V$ is :
$$\begin{matrix} (\textrm A) \, pn - \operatorname{rank} T & (\textrm B) \, mn - p\operatorname{rank} T \\ (\textrm C) \, p(m - \operatorname{rank} T) & (\textrm D) \, p(n - \operatorname{rank} T) \end{matrix}$$
My attempt :
Consider $TX = 0$. Then there are $mp$ homogeneous equations with $np$ unknowns. The dimension of $\textsf V$ will be number of independent entries of the matrix $X$ satisfying $TX=0$. I tried to use rank-nullity theorem of linear transformation but I am not getting any conclusion.
How to proceed from here? Thanks in advance.