Hi, I sort of understand how to prune just number vectors, but these polynomial type questions i am finding a bit tricky, here is one of the questions.

Let $\displaystyle P_{3}(R)$ denote the vector space of real polynomial functions of degree less than or equal to three. Consider the subset $\displaystyle X = \{f1, f2, f3, f4\} \subset P_{3}(R)$ with

$\displaystyle f_{1}(x) = 3x^3 + 2x^2 - x +7$,

$\displaystyle f_{2}(x) = 7x^3 + 5x^2 + 4x + 3$,

$\displaystyle f_{3}(x) = x^3 + x^2 + 6x - 11$,

and $\displaystyle f_{4}(x) = 11x^3 + 8x^2 + x + 2$.

Prune $\displaystyle X$ to produce a linearly independent subset $\displaystyle Y$ such that $\displaystyle Span(Y) = Span(X)$.