Linear Algebra Basics: Finding a Basis for Subspaces in R3"

[Alex6200]

Hi, I had a basic linear algebra question

Question #1

Homework Statement

Find a basis for the subspace of R3 for which the components in all of the vectors sum to zero.

Homework Equations

If u and v are in w and w is a subspace, then a*u + b*v is in w.

The Attempt at a Solution

w = {v in R3 : v1 + v2 + v3 = 0}

Okay, so let's say you have Ax = b, where the column space of A is the basis B, and b is a vector which is in w.

I really don't know how to work with this problem beyond that. I can imagine a basis looking something like:

[1, 0, 0], [0, -1/2, 0], [0, 0, 1/2]

Because if you add those vectors together, all of the components sum to 0. And those are indeed linearly independent. But I don't know if those are the right basis vectors.

Thanks,

Al.

 

[Unco]

Homework Statement

Find a basis for the subspace of R3 for which the components in all of the vectors sum to zero.

Homework Equations

If u and v are in w and w is a subspace, then a*u + b*v is in w.

The Attempt at a Solution

w = {v in R3 : v1 + v2 + v3 = 0}

Okay, so let's say you have Ax = b, where the column space of A is the basis B, and b is a vector which is in w.

I really don't know how to work with this problem beyond that. I can imagine a basis looking something like:

[1, 0, 0], [0, -1/2, 0], [0, 0, 1/2]

Because if you add those vectors together, all of the components sum to 0. And those are indeed linearly independent. But I don't know if those are the right basis vectors. No, a basis of three vectors would span the whole space R3.

Call the subspace described in the problem W.

If v = (x, y, z) is in W, then x+y+z=0. One equation, three unknowns => 2 parameters, so let y=s, z=t. Then we have v = (-y - z, y, z) = (-1, 1, 0)s + (-1, 0, 1)t.

(Note that the condition that x+y+z=0 for each v=(x,y,z) in W is equivalent to saying that W is the perpendicular subspace of span(1, 1, 1).)

 

[HallsofIvy]

Or, slightly different approach, since v1+ v2+ v3= 0, v3= -v1- v2. Let v1= 1, v2= 0 so v3= -1. We have (1, 0, -1). Let v1= 0, v2= 1 so v3= -1. We have (0, 1, -1). Those are basis vectors. That's not the same two vectors as Unco got but there are an infinite number of different bases for this subspace.

 

[Alex6200]

Oh, so when he says "Find a basis", he doesn't mean find all of the bases, he just means find a single vector in the basis?

So if I had another question "Find a basis for a subspace of R3 in which all vectors satisfy:

(1 1 0) v = 0

Then I could just give a vector like:

(-1, 1, 0) and then say that I found a basis?

 

[Office_Shredder]

Oh, so when he says "Find a basis", he doesn't mean find all of the bases, he just means find a single vector in the basis

It means find all of the vectors in a single basis