Final answer:
A basis for the kernel of the transformation L, defined by L(f(x))=f'(−8)−f(1), can be found by determining the set of polynomials p(x)=ax^2+bx+c which satisfy L(p(x))=0. This results in a relationship between the coefficients, yielding a basis of {x^2 - 15x, x}.
Step-by-step explanation:
To find a basis for the kernel of the linear transformation L:P_2[x]\rightarrow\mathbb{R} given by L(f(x))=f'(−8)−f(1), where P_2[x] is the vector space of polynomials in x with degree less than 3, we need to determine the set of all polynomials p(x) such that L(p(x))=0. That is, the derivatives of these polynomials at x=-8 should equal the polynomial evaluated at x=1.
To do so, let's consider a generic polynomial of degree less than 3, p(x) = ax^2 + bx + c. The derivative of p(x) is p'(x) = 2ax + b. Setting up the equation based on the definition of L:
L(p(x)) = p'(-8) - p(1) = -16a + b - (a + b + c)
For L(p(x)) to be zero, we need:
-16a + b - a - b - c = 0
Which simplifies to:
15a + c = 0
This tells us that a and c are not independent. For every choice of a, there must be a corresponding c = -15a to keep the polynomial in the kernel of L. Hence, the polynomial p(x) we are looking for would look like p(x) = ax^2 - 15ax. The coefficient of x (b) can be any real number since it doesn't affect the equation. Thus, one basis for the kernel is {x^2 - 15x, x}.