Find vectors that span a subspace

[csc2iffy]

Homework Statement

Find a pair of vectors that span the subspace x+y-2z=0 of R3.

Homework Equations

x+y-2z=0

The Attempt at a Solution

I just guessed some numbers since its such a simple equation and came up with (1,-1,0) and (2,0,1). I was just wondering what the standard method is to figure this out.. Thanks!

 

[radou]

Well, if a vector is in your subspace, the it must be of the form (2z-y, y, z). If you "extract" y and z as parameters, you have (2z-y, y, z) = y(-1, 1, 0) + z(2, 0, 1). Since (2z-y, y, z) is arbitrarily chosen, it follows that the set {(-1, 1, 0), (2, 0, 1)} spans your subspace, since we have a "rule" on how to obtain the coefficients of the linear combination of these two vectors. Even more, this set is linearly independent, and is actually a basis for your subspace, which is even stronger that what you needed to find.

 

[csc2iffy]

AH! thank you so much! that was driving me crazy!