Question involving linear operators, matrices, and compositions
Upcoming midterms are giving me very little time to work on assignments such as this one, so I need help with this.
Suppose is a vector space over a field .
Suppose is a basis for , is a linear operator with matrix with respect to , is a linear operator with matrix with respect to .
a) Prove that the composition is a linear. (typo?)
b) Prove that the matrix of with respect to is by proving that the th entry of the matrix of is .
A hint provided is as follows: Expand then use this expansion to expand .
This is probably quite easy, but as I said before, I can't devote too much time to this or I'll not be able to properly prepare for my midterms.