# Find bases and kernel of linear map

Find bases for the image and the kernel of the linear map $f:P_2(x) \to P_1(x)$ given by $f(p(x)) = p'(x).$ Based on your results, indicate whether $f$ is injective or surjective.

Ker f = p(x) subset P2 given p(x)=0 = p(x) subset P2 given d/dx p(x)=0 = p0 subset R =R

Img f = q(x) subset P1 given q(x)=d/dx p(x), p(x) subset of P = P1

• ...please...? Some self work, some ideas, some effort...? – DonAntonio Aug 18 '13 at 12:03
• What are P1 and P2? – celtschk Aug 18 '13 at 12:04
• I think $P_2(x)$ means vector spaces of all polynomials of degree atmost $2$ over reals – Marso Aug 18 '13 at 12:08

$\ker f=\{p(x): p'(x)=0\}$ so all constant polynomial. So your $f$ is not Injective.
take any $q(x)\in P_1(x)$ if $f$ is surjective Then there exist $p(x)\in P_2(x)$ such that $f(p(x)=q(x)$ so now can you tell me $f$ is surjective or not?