What does this linear transformation do?

Question

The following is given:

• $F_4$ is the vector space of all symmetrical matrices
• $F = \operatorname{Span}\left\{\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}; \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} ; \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right\}$
• $ L:F \rightarrow F : A \rightarrow A\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}A$

The following is asked:

• What does this linear transformation do ?
• Give an exact description of the kernel and range of the transformation.
• Give the matrix representation M of this transformation
• Eigenvalues and Eigenvectors of the transformation
• What is the dimension of the Eigenspace ?

The kernel, range, matrix representation, eigenvalues and eigenvectors are pretty straightforward. I am struggling with the explanation of what this transformation does and the dimension of the Eigenspace. Can someone explain to me what this transformation does and hint me on the eigenspace question ?

Answer 1

Hint Write $A= \begin{bmatrix} a & b \\ c &d \end{bmatrix}$. Then $$L(A)= \begin{bmatrix} a & b \\ c &d \end{bmatrix}\rightarrow A\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} a & b \\ c &d \end{bmatrix}= \begin{bmatrix} ?? & ?? \\ ?? &??\end{bmatrix}$$

-Kernel: Solve $L(A)=0$.

-Range: describe all values of $L(A)$.

-Matrix Representation: calculate $L$ of the cannonical basis.

-Eigenvalues/Eigenvectors: For which $A$ do you get $L(A)=\lambda A$?