Linear Algebra - Find a subspace

1. Mar 2, 2012

cristina89

1. The problem statement, all variables and given/known data

V is a subspace of the vector space $R^{3}$ given by:
V = {(x, y, z) E $R^{3}$ / x + 2y + z = 0 and -x + 3y + 2z = 0}
Find a subspace W of $R^{3}$ such that $R^{3}$ = V$\oplus$W

I'm really lost in this. My teacher didn't give any example of how to solve this kind of exercise... Can anyone help me how to start and develope this?

2. Mar 2, 2012

lanedance

each of those equations defines a plane - what is the intersection of two planes?

3. Mar 2, 2012

cristina89

A straight line?

Well, if I solve this system:

x = -2y + z
y = -3/5z --> x = -11/5z

Last edited: Mar 2, 2012
4. Mar 2, 2012

Dansuer

To start with, do you know what your looking for ? Do you know what is the direct sum of two subspaces?

5. Mar 2, 2012

lanedance

So the subspace V is a line through the origin, and can be represented by the span of a single vector parallel to that line.

Now onto Dansure's question...

6. Mar 2, 2012

HallsofIvy

W will be a plane perpendicular to that line, containing the origin.