[Math] Dimension of vector space of polynomials

Question

I read on vector spaces lately and then I did a thought experiment. As for a vector in R3 the dimension is 3 (maybe you could give me a proof?), i asked myself what would the dimension be of a vector space of polynomials with degree <= k? I currently think it is infinite as you can always add another polynomial that is not expressible as a linear sum of the set, but I do not have any solid proof for this statement.

Answer 1

Any quadratic polynomial $ax^2+bx+c$ is obviously a linear combination of the three polynomials $x^2$, $x$ and $1$, so that the space of polynomials of degree $\le2$ is at most of dimension $3$.

You can generalize to any degree.