# Span - linear algebra

I'm having some trouble in solving some exercises related to vector spaces, and I can't even start the solution.

I need to check if the sets given span the same subset of the vector space $V$:

(i) $S_1 = ({\sin^2\theta, \cos^2\theta, \sin\theta \cos\theta})$ and $S_2 = ({1, \sin \theta, \cos \theta})$, when $V = C(\mathbb R)$ (all the continuous functions).

(ii) $S_1 = (1,t,t^2,t^3)$ and $S_2 = (1, 1+t,1-t,1-t-t^2)$, when $V = P_3(\mathbb R)$ (all the polynomials with degree $\leq 3$).

In (ii) it is immediate that $1,1+t,1-t,1-t-t^2$ can be written as linear combination of $1,t,t^2,t^3$. In the other direction for example $t=1-(1-t)$ or $t=\frac12(1+t)-\frac12(1-t)$. However, $t^3$ cannot be written as linear combination of $1,1+t,1-t,1-t-t^2$; the reason is that a linear combination cannot have higher degree than the maximal degree of the summands.
In (i) we have for example $\sin^2\theta+\cos^2\theta = 1$, but one needs a somewhat tricky observation to show that the spans differ in the end: We have $f(\theta)=f(\theta+2\pi)$ for all $f\in S_2$; and we have (why?) $f(\theta)=f(\theta+\pi)$ for all $f\in S_1$. As $\sin\theta$ does not have period $\pi$, we conclude that it is not in $S_1$.