Question

Use coordinate vectors to test whether the following set of polynomial span IP'2 . Justify your conclusions. {l - 3t + St 2 , - 3 + St - 7t 2 , -4 +St - 6t 2 , 1 - t 2 }

Answer 1

To test if a set of polynomials spans IP'2, represent each polynomial as a coordinate vector and check if every polynomial in P'2 can be written as a linear combination of the vectors. Determine if a system of equations has a solution for arbitrary values of a, b, and c.

Step-by-step explanation:

To test whether the given set of polynomials spans the vector space P'2, we need to check if every polynomial in P'2 can be written as a linear combination of the given polynomials.

Let's represent each polynomial as a coordinate vector using the standard basis {1, t, t^2}.

The given set of polynomials can be represented as the following coordinate vectors:

{1, -3, 0}, {-3, 1, -7}, {-4, 1, -6}, {1, 0, -1}.

To determine if these vectors span P'2, we need to check if the vector {a, b, c} can be expressed as a linear combination of these vectors. We can set up a system of equations and solve for a, b, and c.

If the system has a solution for any arbitrary values of a, b, and c, then the given set of polynomials spans P'2.