Prove that $(v_n)=(1+t)^n$ is a Basis for $K[t]$.
I cant figure out how to prove this statement. I need to show linear independence. I tried two things:
The first one is with induction: Let $$0=\alpha_1 v_1+\ldots+\alpha_n v_n+\alpha_{n+1}v_{n+1}=\alpha_1 v_1+\ldots+(\alpha_n+\alpha_{n+1}+t \alpha_{n+1})v_{n}.$$ Then we find because of the induction hypothesis that $\alpha_n+\alpha_{n+1}+t \alpha_{n+1}=0$. But we need to conclude that $\alpha_n+1=0$
The second thing I tried is to construct a bijection with the binomial theorem to the standard basis $t^n$. But I only get the map from the basis $(1+t)^n$ to the normal basis $t^n$.
Hint: Show that the map $$K[t]\ \longrightarrow\ K[t]:\ p(t)\ \longmapsto\ p(1+t),$$ is an isomorphism of vector spaces.