# Determine whether or not something is a subspace

1. Mar 7, 2009

### kesun

I guess this kind of topic should belong here. :|

My understanding of the subspace still isn't solid enough, so I want to know what I know so far is at least correct.

By definition, a set of vectors S of Rn is called a subspace of Rn iff for all vectors (I will call them x):
1) (x+y) $$\in$$ S and
2) kx $$\in$$ S.

Also, the solution set of a homogeneous system is always a subspace.

When I encounter problems such as determine whether or not {(x1,x2,x3)|x1x2=0} is a subspace of its corresponding Rn, I would approach this problem as such:

Since the set has only three vectors, then it's in R3, first of all; then I check for 1): Suppose a set of vectors Y {(y1,y2,y3)|y1y2=0}, then (S+Y)=x1x2+y1y2=0+0=0; for 2): kS=(kx1)(kx2)=(k0)(k0)=0. Therefore it is a subspace of R3.

Is my way of solving this problem correct?

2. Mar 7, 2009

### yyat

So far so good...

The equations also need to be linear.

You mean the vectors have three components. The set contains an infinite number of vectors.

No, the set $$S=\{(x_1,x_2,x_3)|x_1x_2=0\}$$ is not a subspace. Try finding $$x,y\in S$$ such that $$x+y$$ is not an element of S.

3. Mar 7, 2009

### kesun

OHHH DAMN! Is it that (x1+y1)(x2+y2) = x1x2+y1x2+x1y2+y1y2? No that definitely ain't the subspace of it..! Am I correct about that?

4. Mar 7, 2009

### Dick

Be concrete. v1=(1,0,0) is in the subspace, and so is v2=(0,1,0). Is v1+v2?

5. Mar 7, 2009

### kesun

Oh, I see. I may need to confirm a few more examples to be sure that I am apply this theorem correctly.

Now suppose {x$$\in$$R5 | ||x||2 $$\geq$$ 0}. First of all, it concerns R5. Since ||x|| is the norm of the vector, which is the same thing as the distance of the vector, so it will always be greater or equal to 0. Since this does not require the outcome to be a specific value(like it did in the previous one, which was 0), is it safe to conclude that this is a subspace of R5?

6. Mar 7, 2009

### Dick

Unless I'm misunderstanding your notation, for ANY vector x in R^5, ||x||^2>=0. So your subspace is ALL of R^5.

7. Mar 7, 2009

### kesun

That's exactly how it stated in this book..So this IS a subspace of R5, right?

8. Mar 7, 2009

### Dick

R^5 is a subspace of R^5. Sure.