I'm not sure what this problem is asking me to do:
Consider the operation M:P2 -> P3 that takes p(x) in P2 to xp(x) in P3. Does this correspond to a linear transformation from R^3 to R^4? if so, what is its matrix?
It's asking you for this. We know that for there is a natural isomorphism between and by and a similar one, call it , between and . If were a linear transformation then so would . So, is a linear transformation?