Linear maps are abstractly defined things. We’d like to make them concrete. We do this by making the following observation: once you know what a linear transformation does on a basis, you know what it ...

which holds purely because composing with an identity map doesn’t change anything. Now apply Theorem 4.19.1 from the previous section twice: you get the change of basis formula: In this subsection ...

Consider $T = (x_1,..,x_5) \to (x_4, x_5)$. We have $\operatorname{null} T = \{(x_1,x_2,x_3,0,0) | x_1,x_2,x_3 \in \mathbb{F}\}$, and $\operatorname{range} T ...