Suppose $f : \mathbb{R}^2 \to \mathbb{R}^2$ is a linear transformation. The two pictures on top in the figure use standard $\mathcal{S}$-coordinates, where $\mathcal{S} = \lbrace \mathbf{e_1}, \mathbf{e_2} \rbrace$. The two figures on bottom in the figure use $\mathcal{B}$-coordinates, where $\mathcal{B} = \lbrace \mathbf{b_1}, \mathbf{b_2} \rbrace$. The figure shows the vectors $\mathbf{b_1}$ and $\mathbf{b_2}$ in blue and the vectors $f(\mathbf{e_1})$ and $f(\mathbf{e_2})$ in red.
 Standard basis $\mathcal{S} = \lbrace \mathbf{e_1}, \mathbf{e_2} \rbrace$ Standard basis $\mathcal{S} = \lbrace \mathbf{e_1}, \mathbf{e_2} \rbrace$ $\Large \stackrel{\displaystyle \lbrack f \rbrack_\mathcal{S}^\mathcal{S}}{\to}$ $\Large \lbrack id \rbrack_\mathcal{B}^\mathcal{S} \ \uparrow$ $\Large \lbrack id \rbrack_\mathcal{B}^\mathcal{S} \ \uparrow$ $\Large \stackrel{\displaystyle \lbrack f \rbrack_\mathcal{B}^\mathcal{B}}{\to}$ Custom basis $\mathcal{B} = \lbrace \mathbf{b_1}, \mathbf{b_2} \rbrace$ Custom basis $\mathcal{B} = \lbrace \mathbf{b_1}, \mathbf{b_2} \rbrace$

1. Find the matrix $A$ for the linear transformation $f$ relative to the standard basis in the domain and in the codomain. That is, find the matrix $A$ such that $\lbrack f \rbrack_\mathcal{S}^\mathcal{S} (\mathbf{x}) = A \mathbf{x}$.
$A =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
2. Write $f(\mathbf{b_1})$ and $f(\mathbf{b_2})$ as linear combinations of the vectors in the basis $\mathcal{B}$. Enter a vector sum of the form 5 b1 + 6 b2.
$f(\mathbf{b_1}) =$
$f(\mathbf{b_2}) =$
3. The vector $\mathbf{b_1}$ an eigenvector for the linear transformation $f$ with eigenvalue (enter a number or DNE).
The vector $\mathbf{b_2}$ an eigenvector for the linear transformation $f$ with eigenvalue (enter a number or DNE).

You can earn partial credit on this problem.