How do I know if it's a subspace of $\mathbb{P}^2$?

Question

$S: (1, x^2, 2+x^2)$. I'm not sure how to test if this set (see left) is a subspace of $\mathbb{P}^2$.The book said that only vectors of the form $s(x^2) +t(1)$ are in spans(S). How did they know that? Also, could anyone recommend any other resources I could use online to help me with Linear Algebra? My textbook leaves a lot of open-ended questions I don't have the concepts down well enough to just figure out myself.

Answer 1

I think you mean to prove $\operatorname{Span}(S)$ is a subspace of $P^2$, in fact, any element $p$ in $\operatorname{Span}(S)$ is $$p=a+bx^2+c(2+x^2)=(a+2c)+(b+c)x^2$$ that is why only vectors of the form $s(x^2)+t(1)$ are in $\operatorname{Span}(S)$.

If you are new to linear algebra, try to prove things by following definition. Like, a subset $W\subset U$ is a subspace of $U$ if and only if $W$ contains zero vector and $u+v\in W, \lambda u\in W$ for $u,v\in W, \lambda\in F$. Try to prove $\operatorname{Span}(S)$ is a subspace of $P^2$ by checking three conditions in the definition.