Basis in the vector space of all polynomials Let $V$ vector space of all polynomials  $p(t) = a_0 + a_1t + \cdots + a_nt^n$,$\forall  n \in\mathbb{N}$ and $a_0,\ldots,a_n \in\mathbb{R}$. How can I prove that $ \gamma = \{1,t,t^2,\ldots\}$ is a basis of $V$, and use it to find a linear transformation $T:V \rightarrow V$ such that $T$ is surjective but not injective. Is $T(x) = x^2$ an example of this transformation? Some help please.

An alternative to the differentiation transformation is the shift,
$T(p(t)) = \dfrac{p(t)-p(0)}{t}$, which sends $a_0+a_1t+\cdots +a_nt^n$ to $a_1+a_2t+\cdots+a_nt^{n-1}$, i.e., it removes the constant term and decreases the other powers by $1$.
For injective but not surjective, there is the shift in the other direction, 
$S(p(t)) = tp(t)$, which increases the powers by $1$.  Or $S(p(t))=p(t^2)$, which doubles all the powers.
