# Problem finding a basis for a given subspace

• Mar 2nd 2010, 06:26 PM
crymorenoobs
Problem finding a basis for a given subspace
Ok, so I need to find a basis for subspace A.

A = {(x1, x2, x3) E R^3 | 2x1 + x3 = 0, x1 + x2 - x3 = 0} of R^3

I understand how to determine whether something is a basis of something else, but I cannot wrap my head around how to determine my own basis for this subspace.
• Mar 2nd 2010, 06:56 PM
TheEmptySet
Quote:

Originally Posted by crymorenoobs
Ok, so I need to find a basis for subspace A.

A = {(x1, x2, x3) E R^3 | 2x1 + x3 = 0, x1 + x2 - x3 = 0} of R^3

I understand how to determine whether something is a basis of something else, but I cannot wrap my head around how to determine my own basis for this subspace.

Set $x_1=t$ then you have a system of two equations in two unknowns.

Using the first equation you get

$x_3=-2t$

Using the 2nd equation we get
$x_2=-x_1+x_3=-t-2t=-3t$

Using this we get

$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} =\begin{bmatrix} t \\ -2t \\ -3t \end{bmatrix}=t\begin{bmatrix} 1\\ -2 \\ -3 \end{bmatrix}$

So any multiple of the last vector will space that space.