Question

Find a basis {p(x),q(x)} for the kernel of the linear transformation

L: P∣x∣→R defined by L(f(x)) = f ′ (−8) − f(1) where P3 [x] is the vector space of polynomials in x with degree less than 3.

Answer 1

To find a basis for the kernel of the linear transformation L, we need to find the polynomials p(x) and q(x) such that L(p(x)) = 0 and L(q(x)) = 0. By setting different values for the constants in the polynomials, we can find p(x) = 1 + x and q(x) = x^2 as the basis for the kernel of L.

Step-by-step explanation:

In order to find a basis for the kernel of the linear transformation L, we need to find the polynomials p(x) and q(x) such that L(p(x)) = 0 and L(q(x)) = 0. For a polynomial f(x) in P3[x], L(f(x)) = f ′ (−8) − f(1).

Let's start by finding p(x).

1. Set f(x) = a + bx + cx^2, where a, b, and c are constants.
2. We have f ′ (−8) − f(1) = (b - 16c) - (a + b + c).
3. Set this equal to 0: (b - 16c) - (a + b + c) = 0.
4. Simplify the equation: -a - 2b - 15c = 0.

By setting b = 1 and c = 0, we get a basis polynomial p(x) = 1 + x. Next, let's find q(x).

1. Set f(x) = a + bx + cx^2, where a, b, and c are constants.
2. We have f ′ (−8) − f(1) = (b - 16c) - (a + b + c).
3. Set this equal to 0: (b - 16c) - (a + b + c) = 0.
4. Simplify the equation: -a - 2b - 15c = 0.

By setting b = 0 and c = 1, we get a basis polynomial q(x) = x^2. Therefore, the basis for the kernel of the linear transformation L is {p(x) = 1 + x, q(x) = x^2}.