Let $B : F^2 \to F^2$ be a linear transformation with distinct eigenvalues $y_1,y_2$ and corresponding eigenvectors $v_1,v_2$. Find three bases in which $B$ is diagonal. Write each basis, and the matrix for $B$ in that basis.
May I get some help for this question? I'm not sure how to start :/
For a linear transformation, the whole beauty of having as many eigenvectors as the dimension of the vector space is that they allow us to diagonalize the matrix of the transformation. Recall that to construct the matrix $M_T$ of a linear transformation $T$ with respect to a basis, we apply the transformation to all basis elements (in the order they are listed in the chosen basis) to obtain the columns of $M_T$. For example, consider $\{v_1,v_2\}$ as a basis for $F^2$. Since $$Bv_1=y_1v_1=y_1v_1+0v_2 \quad \text{and} \quad Bv_2=y_2v_2=0v_1+y_2v_2,$$ the corresponding matrix is $$M_B=\begin{bmatrix} y_1 & 0 \\ 0 & y_2\end{bmatrix}.$$ Now, can you think of other ways of creating a basis using $v_1$ and $v_2$ or their multiples (because they will be eigenvectors as well), and not necessarily in this order?