[x,1] \text{ for } P_1 \text{ and } [(1,0),(0,1)] \text{ for } \mathbb R^2. When we have fixed a basis for a vector space, @def-basis together with @thm-linear-indep-defs and @def-span tell us that, ...

\text{Property 1:} \quad & L(a x )= a L(x) \\ \text{Property 2:} \quad & L( x + y ) = L(x) + L(y). \end{aligned} Let's list a few things that must be true to arrive ...