I have an exam coming up and I have a whole bunch of questions to study from, and I am realizing that I really don't have a good grasp on basis matrices for linear transformations. One of the questions is:
For a linear map $f$ in $Hom_F( V,W)$ the (operator) adjoint $f^*:W^*\rightarrow V^*$ is defined by $f^*(g)=g\circ f$. If $f$ has matrix $A$ in some basis, what is the matrix of $f^*$ in the dual basis?
Similarly, here is another question that Im struggling a bit with that would help if I could get some intuition on...
A linear map between finite dimensional vector spaces $f:V\rightarrow W$ corresponds to a matrix when bases for $V$ and $W$ are chosen. HOw does this matrix change if we change bases in $V$ and $W$.