What kind of set would generate $R^n$ ? What about $P_n$ ? Also, the exam question that made me ask this:
Does the set $S = \{1, x^2, x^2 + 2\}$ generates $P_2$ ?
I'm a little bit uncertain about when does a set generates a vector space, therefore I can't figure out if it does or not.
No, the set $S$ does not generate $P_n$. The lack of an $x$ term from any of the polynomials in $S$ means that polynomials such as $x + 1$ and $x^2 + 2x + 1$ cannot be linear combinations of polynomials in $S$.