Finding matrix relative to a polynomial basis and basis in R2?

Question

Let U be the space of polynomials with basis E=[1,t,t^2], let V=R2 have the basis F=[(1,0),(1,1)] and define a linear map T: U->V by T(f)= (f(3),f'(3)).

1. Determine the matrix representing T relative to the basis E and F.

2. What is the Rank of T?

3. Exhibit a basis for the kernel of T.

I understand how to determine a matrix relative to a single basis in R2, but I've never done it relative to two basis, and never with a polynomial as a basis. Can anyone walk me though how to go about solving this type of problem?

Answer 1

HINT

Polynomial of degree 2 in this case are a vector space in the sense that at each polynomial $f(t)=a+bt+ct^2$ we can associate the vector $v=(a,b,c)$.

The linear map $T$ maps a polynomial $f(t)=a+bt+ct^2$ in $(a+3b+9c,b+6c)$. Since we are requeste to use the basis F we need to write $(a+3b+9c,b+6c)$ in this basis that is

• $x+y=a+3b+9c\implies x=a+3b+9c-b-6c=a+2b+3c$
• $y= b+6c$

Then

$$T(a,b,c)=(a+2b+3c,b+6c)$$

Can you proceed from here?