i just started on diagonalizable matrix and i am trying to prove that f: 2x2 matrix--> 2x2 matric where f(A)= A + A^t. find a basis for the 2x2 matrix which consist of eigenvectors for f.

im wondering, the basis would be in the form of a 2x2 matrix yet eigenvectors are column matrices..so how do i go about doing it when the sizes are different?