Canonical Form of Linear Map

Question

From Lee's Smooth Manifolds Appendix

I have stared at this for some time and I just don't quite see how this follows.

Answer 1

Remember you are picking one bases for $V$, and a different one for $W$.

Let's rename the basis of $V$ to $\{D_1,\dots, D_{r+k}\}$.

For $1 \leq i \leq r,\ T(E_i) = F_i$ and for $1 \leq j \leq k,\ T(K_j) = 0$, so $T(D_i) = \begin{cases} F_i\text{ if }i\leq r\\0\text{ otherwise}\end{cases}$

What you write in the $i-th$ column of the matrix representation of $T$ is the coordinates of $T(D_i)$ in the basis you picked for $W$, so the conclusion follows.

Let me know if you want more details (do you know what 'coordinate' means?)