Finding a basis for a subspace

[SOLVED] Finding a basis for a subspace

Homework Statement

We have a subspace U in R^3 defined by:

U = {(x_1 , x_2 ; x_3) | x_1 + 2*x_2 + x_3 = 0 }.

Find a basis for U.

The Attempt at a Solution

We have the following homogeneous system:

(1 2 1 | 0).

From this I find the solution to be written as a*(2,1,0)^T + b(-1,0,1)^T, where a and b are arbitrary constants. From this I find three linearly independant vectors.

Am I correct?

(2,1,0) isn't a solution (but it's close). Also, if your subspace contains 3 linearly independent vectors then it's all of R^3. This is obviously not the case right?

The solution was (-2,1,0).

Yeah, I need a condition too so it is just the subspace where x_1 + 2*x_2 + x_3 = 0 - but how do I include that?

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Well, in the answer you gave I see two vectors (one with the correction you mentioned) that appear to be linearly independent. They must be a basis of the solution space, right?

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Dick