Show that the mapping
$g: (u,v)$ $\longmapsto$ $(sinuv + u, u+v, uv)$
maps $R^2$ onto S, with S be the surface defined by the equation $z = (x- sinz)(sinz-x+y)$
Proof:
Since $g(1,0) = (1,1,0)$ and $g(0,1) = (0,1,0)$ and these two vectors are linearly independent, they are the basis for S.
I don't understand how this means that every $(x,y,z)$ in S can be written as a linear combination of $(1,1,0)$ and $(0,1,0)$
We have just to show that
$u v=(\sin (u v)-\sin (u v)+u) (-(\sin (u v)+u)+\sin (u v)+u+v)$
indeed RHS simplifies to
$u(-\sin (u v)-u +\sin (u v)+u+v)=uv$
So for any $(u,v)\in\mathbb{R}^2$ the image is a point of $S$
Hope this is useful