Question

The set b={2-3x², 10−x+15x², 3x−26−42x²} is a basis for p₂. find the coordinates of p(x)=2x−14−24x² relative to this basis:

[p(x)]B= [ ___ ]
[ ___ ]
[ ___ ]

Answer 1

To find the coordinates of a polynomial relative to a given basis, you express the polynomial as a linear combination of the basis polynomials and solve the resulting system of equations for the coefficients, which are the coordinates.

Step-by-step explanation:

To find the coordinates of p(x) = 2x - 14 - 24x² relative to the basis B = \{2-3x², 10-x+15x², 3x-26-42x²\}, we need to express p(x) as a linear combination of the basis vectors. That means finding scalars a, b, and c such that:

p(x) = a(2-3x²) + b(10-x+15x²) + c(3x-26-42x²).

Comparing coefficients, we get the following system of equations:

For the coefficient of x²: -24 = -3a + 15b - 42c,

For the coefficient of x: 2 = -b + 3c,

For the constant term: -14 = 2a + 10b - 26c.

Solving this system, we find the values for a, b, and c that represent the coordinates of p(x) relative to the basis B. These coordinates are the entries for [p(x)]_B.

In the context of this problem, we have not solved the actual system of equations but would proceed using methods such as substitution or matrix operations to find the values of a, b, and c.