Basis Change and Matrix Similarity

Every subspace U C Rn that differs from the trivial subspace 0 has many bases. 

One basis of R is the standard unit vector basis = ei, e2, e . Since 

Note that both U and V are invertible, or nonsingular since the and the Vj are both linearly independent as bases. Thus 

How does a basis change in R alter the matrix representation of a linear map / Rn - Rm  

In subsection (A) we have expressed a given linear transformation / as a matrix A with respect to the standard unit vector basis = e, e2, e , namely 

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