Unique Subspaces for Vector Space V in R3

[muzihc]

Homework Statement

If R3 is a vector space and V = (x,x,0) is a subspace, find unique subspaces W1 and W2 such that R3 = V ⨁ W1 = V ⨁ W2

Homework Equations

The Attempt at a Solution

Assuming R3 = (x,y,z) - please correct me if I'm wrong somehow - then I could pick a W1 like (0,-x+y,z), but then that seems to leave me no room to pick a distinct W2.

I might be missing something very basic, I'm not sure. I've spent plenty of time thinking about this.

 

[Phrak]

W1 could be a plane in R^3 not containing (1,1,0). W2 could be another plane.

 

[muzihc]

Thanks.

In general, how would I define a plane in R^3?

 

[chiro]

Thanks.

In general, how would I define a plane in R^3?

A line in some space has one independent dimension. A plane has two.

The equation of a plane in R3 is of the form:

ax + by + cz + d = 0, where (a,b,c,d) are constants.

So by having two free parameters (say u and v) you can write the other in terms of those two.

Say let's say your plane equation is z = u and y = v for (a,b,c,d) = (1,1,1,-1) you

have

x + y + z - 1 = 0

So z = u, using standard basis [0,0,1] for z vector and y = v for y vector [0,1,0] you get

x = 1 - y - z = 1 - u - v

So you would have the system:

(1 - u - v)*[1 0 0]^T + u * [0 1 0]^T + v*[0 0 1]^T

= [1 0 0]^T + u*[-1 1 0]^T + v*[-1 0 1]^T

 

[Phrak]

You don't want to define a plane, but the bases vectors of W1 and W2. Think of a couple planes that include the z component. Remember the basis vector (1,1,0) doesn't have a z component, so to cover all of R^3, W1 and W2 have to include it.