Final answer:
To test the linear independence of a set of polynomials, we can use coordinate vectors. By setting up a system of equations with the coordinate vectors and solving for the coefficients, we can determine if the set of polynomials is linearly dependent or independent.
Step-by-step explanation:
To test the linear independence of a set of polynomials, we can use coordinate vectors. In this case, we will consider the set of polynomials {1 - 2t² - t³, t + 2t³, 1 + t - t²}. To test for linear independence, we need to determine if there exists a non-zero solution to the equation a(1 - 2t² - t³) + b(t + 2t³) + c(1 + t - t²) = 0.
We can express the polynomials as coordinate vectors, where each coefficient represents a coordinate. For example, the first polynomial can be expressed as (1, 0, -2, -1). By setting up a system of equations with the coordinate vectors and solving for a, b, and c, we can determine if the set of polynomials is linearly independent or dependent.
If the system of equations has a unique solution where a = 0, b = 0, and c = 0, then the set of polynomials is linearly independent. However, if the system of equations has infinitely many solutions, with at least one solution where a ≠ 0, b ≠ 0, or c ≠ 0, then the set of polynomials is linearly dependent. In this case, the non-zero solution would serve as the linear dependence relation.