Linear Algebra: Deriving a span from an equation

[turbowaffle]

I can intuitively see that for
y=9x, the vector [9,1] spans the subspace created by that equation. I'm not sure how to actually derive it though, for say

x+3y-2z=0

It seems simple but I'm not sure how to determine which vectors span the subspace of the equation.

 

[pka]

for y=9x, the vector [9,1] spans the subspace created by that equation. [But]x+3y-2z=0 to determine which vectors span the subspace of the equation.

In the first place, I hope you mean [1,9] spans the one-dimensional subspace. Any scalar multiple of [1,9] is a point on the line.

Now x+3y-2z=0 is a two-dimensional subspace so we need two linearly independent spanning vectors for that plane. Because x=-3y+2z, we can choose [-3,1,0] & [2,0,1].

 

[turbowaffle]

Ok, that makes sense. I thought it had something to do with finding some vectors in the image but I wasn't quite sure how to do it, though now it seems obvious. Thanks!