find the kernel of the linear transformations T: P 2 to R given by T(p(x))=p\'\'

Question

find the kernel of the linear transformations

T: P2 to R given by T(p(x))=p'''(x).

Explanation / Answer

The kernel of a linear transformation is defined as, If a linear transformation is from V->W such that if u is a vector in V such that T(u)=0, then vector u is the kernel of T. So, if we apply this case, then T(p(x))=p'''(x) the kernel of T is all sets of p(x) that p'''(x)=0. So the kernel of T is all the homogeneous solutions to the equation. Since the derivative is a simpler form, we can evaluate the derivative here. Given that p'''(x)=0, then p''(x)=C1, where C1 is some constant p'(x)=C1x+C2, where C1 and C2 are some constants p(x)=C1/2*x+C2x+C3, where C1, C2, and C3 are some constants. With given initial conditions, we can find the solution for p(x), which is the kernel for T

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