Let W be the space of all n X 1 column matrices over a field F. If A is an
n X n matrix over F, then A defines a linear operator L on W through left
multiplication: L(X) = AX. Prove that every linear operator on W is left multiplication
by some n X n matrix, i.e., is L for some A.
Now suppose V is an n-dimensional vector space over the field F, and let b be an ordered basis for V. For each a in V, define Ua = [a]b (a in b-coordinates). Prove that U is an isomorphism of V onto W. 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?
Any help on these two problems would be appreciated.