TITLE: Translating an Italian exercise precisely
QUESTION [8 upvotes]: I am trying to help a friend with his algebra course.  However, his exercises are in Italian, and unfortunately he translates them poorly for me since he does not know the mathematical terms in English. I have tried to translate it myself, but without luck. I still do not know what exactly to do.
More precisely, it is the description in the exercise and exercise (a) that I do not understand completely; the rest I understand.


E2) Sia $f\colon\mathbb{C}^4\to\mathbb{C}^4$ una trasformazione lineare e si supponga che la matrice associata a $f$ rispetto alla base $\mathcal{B} = \{\mathbf{e}_2; \mathbf{e}_1; \mathbf{e}_3+\mathbf{e}_4; \mathbf{e}_3+\mathbf{e}_2\}$ su dominio e codominio ($\mathbf{e}_i$ sono i vettori della base canonica di $\mathbb{C}^4$) sia
$$\mathbf{A} = \begin{bmatrix}3&3&2&2\\ 3&3&2&2\\ 0&0&1&1\\ 0&0&1&1\end{bmatrix}$$
(a) Si determini la matrice $\mathbf{B}$ associata a $f$ rispetto alle basi canoniche.
(b) Si calcoli la dimensione dell'immagine di $f$.
(c) Si dica se la matrice $\mathbf{B}$ è diagonalizzabile.
(d) Si calcoli una base dello spazio nullo dell'applicazione lineare $f$.

REPLY [9 votes]: Let $f:\mathbb{C}^4 \rightarrow \mathbb{C}^4$ be a linear map and suppose that the matrix associated with $f$ with respect to the basis $\mathcal{B}=\{ e_2,e_1,e_3 + e_4, e_3 + e_2\}$ (where $e_i$ are the canonical basis vectors of $\mathbb{C}^4$) is
(the matrix $A$ written above in the exercise).
(a) Determine the matrix $B$ associated with $f$ with respect to the canonical basis of $\mathcal{C}^{4}$.
(b) Calculate the dimension of the image of $f$.
(c) Say whether the matrix $B$ is diagonalizable.
(d) Calculate a basis of the null space of the linear map $f$.