Let $R=k[x_0,x_1]$ and $R'=k[y_0,y_1]$ be two graded rings. We define a map $$k[u_{00},u_{01},u_{10},u_{11}]\rightarrow \bigoplus_{n\geq0}(R_n\otimes R_n')$$ given by $u_{ij}\rightarrow x_i\otimes y_j$. How to show that the kernel is $(u_{00}u_{11}-u_{01}u_{10})$?
I want to use this to prove that $\mathbb{P}_k^1\times\mathbb{P}_k^1$ can be embedd into $\mathbb{P}_k^3$ as a quadratic surface.
Any proof or reference is appreciated.