Translating an Italian exercise precisely

Question

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 $f$.

Answer 1

Suppose $f$ is a linear transformation and suppose further that the matrix of $f$ with respect to the basis $B = \{ e_2; e_1; e_3 + e_4; e_3 + e_2 \}$ is the matrix $A = ...$, where the basis $B$ is used as the basis of both the domain and codomain ($e_i$ being the vectors of the standard basis of $\mathbb{C}^4$)

a. Determine the matrix for the transformation $f$ with respect to the canonical (standard) basis.

b. Calculate the dimension of the image of $f$ (i.e., the "rank" of $f$).

c. Say whether the matrix $B$ for $f$ is diagonalizable.

d. Calculate a basis for the null-space (or kernel) of the linear transformation $f$.

There you go. That's a pretty solid translation of the exercise. Now, perhaps., you can confidently work through it.

Answer 2

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 $f$.