I shoud show whether $\displaystyle A: V \rightarrow \mathbb{F}^m$ and for all $\displaystyle A(v)=\begin{bmatrix} f_1(v)\\ f_2(v)\\ \vdots\\ f_m(v)\end{bmatrix}$ and $\displaystyle f_i$ is a linear functional $\displaystyle f_i \in L(V, \mathbb{F}^1), i=1,2,\dots ,m$. is a linear transformation. I think no, because the image of the zero vector is not 0. To be honest I am not really sure about these lienar functional thing.

Second. $\displaystyle C \in L(V), \{v_1,v_2,v_3\}$ v1 v2 v3 is a basis of V.

$\displaystyle C(\alpha v_1+\alpha v_2+\alpha v_2)=(\alpha-\beta+2\gamma)v_1 + (2\alpha + \gamma)v_2 + (3\alpha + \beta)v_3$. About this I know it is a linear transformation. Is it invertible. I know it would be enough to show its injectivity but how can I do this in this case?

Thank you!