I am currently looking at a set of polynomials that looks like this:
$$S = \{p_0 + p_{1}x + p_{2}x^2 + p_{3}x^3 \in P_3 | p_0 = p_1 = 5p_3\} $$
$$ V = p_3$$
V is a vector space and S represents a subspace of V. How can I go about determining the finite spanning set of this? I'm been re-reading my textbook for over an hour and I am just so confused!
$p_0=p_1=5p_3\implies p_0+p_1x+p_2x^2+p_3x^3=p_0(1+x+\dfrac{x^3}5)+p_2x^2$. Thus, $S$ is spanned by the vectors $1+x+\dfrac{x^3}5,x^2$.