Suppose $V$ and $W$ are finite-dimensional. Let $v \in V$. Let

1. Show that $E$ is a subspace of $\mathcal L(V,W)$.
2. Suppose $v \ne 0$. What is $\dim E$?

Obviously $(T_1+T_2)v = T_1v+T_2v = 0$ so $E$ is closed under addition, likewise $(\lambda T)v = \lambda(Tv) = 0$ making it closed under addition and scalar multiplication, hence it's a subspace. This shows (1)

For (2) let $v_1 = v$ and extend to a basis $v_1,\dots,v_n$ of $V$. Pick any basis $w_1,\dots,w_m$ of $W$.
Because $\mathcal M(Tv_1) = \mathcal M(T)\mathcal M(v_1) = 0$ the first column of $\mathcal M(T)$ must be zero (since multiplying by $v_1$ just extracts the first column)
The other columns can be anything, therefor $\dim E = (n-1)m = (\dim V - 1)(\dim W)$ (simply count the parameters that can be anything)

todo: make rigorous, cite theorems from 3.d