TITLE: Vectors spanning a plane
QUESTION [6 upvotes]: I am having a problem with this question, I just can't seem to get it.
Consider the plane $\mathbb{R}^3$ defined by the equation $x+2y-z=0$ Find any two vectors $\mathbf{v},\mathbf{w}$ such that their span may be identified with this plane.

REPLY [10 votes]: Let
$M=\{(x,y,z)^T\in \mathbb{R^3}:x+2y-z=0\}$
$x+2y-z=0$$\hspace{0.2cm}$$\implies$$\hspace{0.2cm}$
$z=x+2y$
So
$(x,y,z)^T=(x,y,x+2y)^T=x(1,0,1)^T+y(0,1,2)^T$
Hence 
$M=\{(x,y,z)^T\in \mathbb{R^3}:x+2y-z=0\}=<(1,0,1)^T,(0,1,2)^T>$