Question

Find a polynomial p₃ such that {p₀, p₁, p₂, p₃} is an orthogonal basis for the subspace P₃ of P₄. Scale the polynomial p₃ so that its vector of values is (-1,2,0,-2,1).

Answer 1

To find the polynomial p₃ that forms an orthogonal basis for the P₃ subspace of P₄, we analyze the dimensions and assign the given values to the polynomial coefficients.

Step-by-step explanation:

To find a polynomial using the orthogonal basis {p0, p1, p2, p3} for the subspace P3 of P4, we start by understanding the dimensions of the polynomials. P3 represents polynomials of degree 3 or less, while P4 represents polynomials of degree 4 or less.

Since the polynomial p3 belongs to P4, it has 5 coefficients. Since the given vector of values is (-1, 2, 0, -2, 1), we assign these values to the respective coefficients of the polynomial in descending order of degree:

1. coefficient of x4: -1
2. coefficient of x3: 2
3. coefficient of x2: 0
4. coefficient of x: -2
5. coefficient of constant term: 1

Therefore, the polynomial p3 can be written as:

p3(x) = -x4 + 2x3 - 2x + 1