Let's $S=k[x_0,x_1,x_2]$ be a standard graded ploynomial ring and $k$ be a alg. closed field. Consider the following Veronese map $\nu_4:\mathbb{P}^2=\mathbb{P}(S_1)\to\mathbb{P}(S_4)=\mathbb{P}^{14}$. I know that the spanning of six points will give us a $6-dimensional$ vector space and then we have $\mathbb{P}^5$. Suppose that $\{[L_1^4],...,[L_6^4]\}$ be a set of six generic points in $\mathbb{P}^{14}$.
I am going to show that $$Span([L_1^4],...,[L_6^4])=<[L_1^4],...,[L_6^4]>_k=\mathbb{P}^5$$ Thank you for your helps.