Let $V$ and $W$ be finite dimensional vector spaces over a field $F$ with $\dim_F(V ) = \dim_F(W )$ and let $T : V → W$ be a linear map. Prove there exists an ordered basis $A$ for $V$ and an ordered basis $B$ for $W$ such that $[T ]_B^A$ is a diagonal matrix where every entry along the diagonal is either a $0$ or a $1$.
I understand that I need to prove this by Rank-Nullity Thrm and but I am confused on how to apply it? Pls help me out.
Let $V'$ be a subspace of $V$ such that $V=\ker(T)\oplus V'$. Let $(a_1,\ldots,a_k)$ be a basis of $V'$ and let $(a_{k+1},\ldots,a_n)$ be a basis of $\ker(T)$. For each $j\in\{1,2,\ldots,k\}$, let $b_j=T(a_j)$. Add to $b_1,\ldots,b_k$ linearly independent vectors $b_{k+1},\ldots,b_n$ such that $(b_1,\ldots,b_n)$ is a basis of $W$. Finally, define $A=(a_1,\ldots,a_n)$ and $W=(b_1,\ldots,b_n)$. Since you have$$T(a_j)=\begin{cases}b_j&\text{ if }j\leqslant k\\0&\text{ otherwise,}\end{cases}$$$A$ and $B$ are the bases that you're after.