I shoud show whether A: V \rightarrow \mathbb{F}^m and for all A(v)=\begin{bmatrix} f_1(v)\\ f_2(v)\\ \vdots\\ f_m(v)\end{bmatrix} and f_i is a linear functional 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. C \in L(V),  \{v_1,v_2,v_3\} v1 v2 v3 is a basis of V.
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!