Thread: Beginning vector subspace question

"Determine if the given set is a subspace of Pn for an appropriate value of n. Justify your answers."

The first problem under these directions is -

"All polynomials of the form p(t) = at^2, where a is in R."

I know what the directions say...but I still don't really get exactly what it's asking. The back of the book says "Yes, by Theorem 1, because the set is Span {t^2}."
Theorem 1 - "If v1, ... , vp are in a vector space V, then Span {v1, ... , vp} is a subspace of V."

I don't get how this proves the above problem. To determine if something is a subspace, shouldn't I be given a vector? Or set of vectors? How does a polynomial with degree 2 fit into that theorem? Can anyone please explain what I should be trying to find in each problem like the one above or how to go about trying to find it? Any help is appreciated, thanks.

There are several conditions for something to be a space. For instance,
if A and B are in the space, is A+B in the space. Also, if (e.g.) A is in the space, is 3A in the space?

To start off, for your case, if p(t) is at^2, and q(t) is bt^2, is p(t)+q(t) = ct^2 for some c?

So...is t supposed to be treated as a vector and t's coefficient a scalar?

You can think of it that way. Alternatively, you can think of t^2 as just something fixed. It really depends on how the problem is set up. From the answer in the book, it doesn't seem like they want t^2 to be variable.