Question

For the vector space of linear functions P₁, we will use the standard basis polunomials: {1,x}. Recall that a polunomial such as p(x) = 7 + 6x would be the vector [7 6] in this space.

The function F, defined by F(p(x)) = ⁴∫₀ p(x)dx, is a linear transformation from P₁ to R.

Write the matrix for this intrgral linear transformation according to the standard basis polynomials.

Answer 1

The matrix for the linear transformation F(p(x)) = ⁴∫₀ p(x)dx in the vector space P₁, with the standard basis {1, x}, is [4 8]. This represents the integration of the constant term and the x-term of any polynomial in P₁ over the interval [0, 4].

Step-by-step explanation:

The student has asked about the matrix of a linear transformation that takes a polynomial p(x) in the vector space P₁ and maps it to a real number through integration. The transformation is defined as F(p(x)) = ⁴∫₀ p(x)dx. To find this matrix with respect to the standard basis {1, x} for P₁, we evaluate F on each basis element.

First, for the constant polynomial 1, we have:

F(1) = ⁴∫₀ 1 dx = x ∫⁴₀ = 4 - 0 = 4

Then, for the polynomial x, we have:

F(x) = ⁴∫₀ x dx = \frac{1}{2} x^2 ∫⁴₀ = \frac{1}{2} (16 - 0) = 8

Therefore, the matrix for F with respect to the basis {1, x} is:

[4 8]

This matrix shows that the linear transformation of any polynomial p(x) = a + bx in P₁ is a 4-to-1 mapping to the real numbers, where the number 4 is the result of integrating the constant term, and the number 8 results from integrating the x-term over the interval from 0 to 4.