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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.

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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

$M=\{(x,y,z)^T\in \mathbb{R^3}:x+2y-z=0\}=<(1,0,1)^T,(0,1,2)^T>$

Just take two vectors that satisfy the equation of the plane for example

$$v=(-1,1,1)^T\quad;\quad w=(0,1,2)^T$$

and verify that they are linearly independent i.e. there’s not $k\in\Bbb R$ such that

$$v=kw$$

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