The polynomial p3 can be written as: p3(x) = -x4 + 2x3 - 2x + 1
To construct a polynomial for the subspace P3 of P4 using the orthogonal basis p0, p1, p2, p3, we must first understand the dimensions of the polynomials. P3 stands for polynomials of degree 3 or less, and P4 stands for polynomials of degree 4 or less.
Because p3 is a polynomial belonging to P4, it has 5 coefficients. Because the given vector of values is (-1, 2, 0, -2, 1), we assign these values to the polynomial coefficients in descending order of degree:
coefficient of x4: -1
coefficient of x3: 2
coefficient of x2: 0
coefficient of x: -2
coefficient of constant term: 1
Therefore, the polynomial p3 can be written as: p3(x) = -x4 + 2x3 - 2x + 1.
Complete question:
Find a polynomial P₃ such that {po, P₁, P₂, P₃} (see Ex- ercise 11) 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).