The standard basis and two custom bases and for are shown in the figures below. Suppose that is the linear transformation defined by relative to the standard basis in the domain and the standard basis in the codomain.
Standard basis
Standard basis
Custom basis
Custom basis
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 .
Find the change of basis matrix from -coordinates to standard -coordinates. That is, find the matrix such that .
Find the change of basis matrix from -coordinates to standard -coordinates. That is, find the matrix such that .
Find the matrix for the linear transformation relative to the basis in the domain and in the codomain. That is, find the matrix such that .
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 .
Find the matrix for the linear transformation relative to the basis in the domain and in the codomain. That is, find the matrix such that .
You can earn partial credit on this problem.