L P3 P3 Defined by Lpx xpx with respect to the basis 1xx2

L : P³ -> P³

Defined by L(p(x)) = xp\'(x) with respect to the basis {1,x,x²}

Extra Credit: What is the range of the linear transformation in problem 3?

Solution

Suppose that p(x) = ax² + bx + c is in the kernel of L.

So, L (p(x)) = 2ax² + bx

Thus, if p(x) is in the kernel of L, 2ax² +bx = 0 for all x, which implies that a = 0 and b = 0. Thus, every polynomial in the kernel of L is of the form p(x) = c.

Therefore,ker(L) = Span(1)

To determine the range of L, we again consider an arbitrary polynomial p(x) = ax²+bx+c, and apply L to the polynomial L (p(x)) = 2ax²+bx.

Thus, the range of L is all polynomials of the form 2ax² + bx.

Hence, range(L) = Span(x, x² )