 Solution

Use change of basis matrices whose columns are the respective basis vectors in standard coordinates:The coordinate relation is $[\Phi (x){]}_C=A[x{]}_B$. Therefore the standard-coordinate matrix of a linear map is $T=S_{C}A{S}_B^{-1}$, and the new coordinate matrix is $A^{\prime }=S_{C^{\prime }}^{-1}T{S}_{B^{\prime }}$. Multiplying givesOne can also read off the columns: the two new domain basis vectors map to twice the first and four times the second new codomain basis vector. All four change of basis matrices are invertible. In particular the printed codomain vectors are three-dimensional; retaining their third coordinates is essential.