Suppose is a linear transformation. The two pictures on top in the figure use standard -coordinates, where . The two figures on bottom in the figure use -coordinates, where . The figure shows the vectors and in blue and the vectors and in red.

Standard basis

Standard basis

Custom basis

Custom basis

Part 1: Eigenvectors and eigenvalues

Find the matrix for the linear transformation relative to the standard basis in the domain and in the codomain. That is, find the matrix such that .

Write and as linear combinations of the vectors in the basis . Enter a vector sum of the form 5 b1 + 6 b2.