Suppose V is finite$-$dimensional and v$1$; : : : ; vm $2$ V$.$ Define a linear

map W V $0!$ Fm by

$.\text{'}/$ D

$\text{'}.$v$1/$; : : : ; $\text{'}.$vm$/$

:

$($a$)$ Prove that v$1$; : : : ; vm spans V if and only if is injective.

$($b$)$ Prove that v$1$; : : : ; vm is linearly independent if and only if is

surjective.