Exercise 2.4.12

We can check ${\varphi }_{\beta }$ is linear first. For $x={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^n{a}_i{v}_i$ and $y={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^n{b}_i{v}_i$, where

we have that

And we can check whether it is injective and surjective. If ${\varphi }_{\beta }(x)=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill ⋮\hfill \\ \hfill 0\hfill \end{array}\right)$then this means $x={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^{n}0v_i=0$. And for every $\left(\begin{array}{c}\hfill a_1\hfill \\ \hfill a_2\hfill \\ \hfill ⋮\hfill \\ \hfill a_n\hfill \end{array}\right)+c$ in ${𝔽}^n$, we have that $x={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^n{a}_i{v}_i$ will be associated to it.