You've lost me here. You say "let a be an ordered basis" but then say "for each a in V". Did you mean "let b be an ordered basis"? Does "[a]b" mean "the coefficients when a is written as a linear combination in that basis"? (I was try to think of "b" as a vector!) "W" is still the column matrices, right? Remember the basis I suggested for W above? Do you see that U maps each vector of b into the corresponding member of that basis. And any linear transformation that maps basis vector to basis vectors is an isomorphism.
Now suppose V is an n-dimensional vector space over the field F, and let a
be an ordered basis for V. For each a in V, define Ua = [a]b. Prove that U is an
isomorphism of V onto W.
Basically, what I said before. A is the matrix representing T in this basis. Its columns are the coefficients in the linear combination of each
If T is a linear operator on V, then UTU^(-1) is a linear
operator on W. Accordingly, UTU^(-1) is left multiplication by some n X n matrix A.
What is A?