# Homework Help: Figuring out if subset of R2 is a subspace

1. Dec 14, 2011

### csc2iffy

1. The problem statement, all variables and given/known data
Let E be the subset of R2 defined by E={(x,y)|x≥0,y$\in$ℝ}. Is E a subspace of R2?

2. Relevant equations
E={(x,y)|x≥0,y$\in$ℝ}

3. The attempt at a solution

2. Dec 14, 2011

What are the conditions for a space to be a subspace of some other space? Can you apply these to find a counterexample, perhaps?

3. Dec 14, 2011

### csc2iffy

ahh.. So E is not closed under scalar multiplication since -k(x,y)=(-kx, -ky) and this is not an element of E

4. Dec 14, 2011

Exactly. Note only that you have to take an element (x, y) of E where x > 0 to demonstrate that it works, since if you take an element of the form (0, y) in E, then for some k < 0 you have k(0, y) = (0, ky), which is still in E.

5. Dec 14, 2011

### csc2iffy

Thanks! I was also struggling a bit with this problem:
Let E={(x,2x+1|x∈ℝ}, so E is a subset of R2. Is E a subspace of R2?

I said no because it is not closed under addition:
(x,2x+1)+(y,2y+1)=(x+y,2x+1+2y+1)=(x+y,2(x+y)+2) which isn't equal to (x+y,2(x+y)+1)
Is this correct?

6. Dec 14, 2011

Yes. Also, it's even easier to see that (0, 0) is not in E in this case.

7. Dec 14, 2011

### csc2iffy

Oh that is much easier! Thank you so much!

8. Dec 14, 2011

### csc2iffy

Sorry, no one is replying to my other thread so I figured I would ask you one more thing here.

Let E={(2a,a)|a∈ℝ}. Let B={(b,b)|b∈ℝ}.
Is E∪B a subspace of R2?
What is E+B

My solution:
E∪B={(2a,a),(b,b)|a,b∈ℝ}
I don't know how to show if tis is a subspace of R2 or if that is the correct union

For the second part, I know that E+B = span (E∪B)
but I don't know if I have the right union?

9. Dec 14, 2011

A union of subspaces of a given space need not be a subspace of that space. For example, take a non-zero a in R, and let (2a, a) and (a, a) be elements of E U B. Then (2a, a) + (a, a) = (3a, 2a) is neither in E nor in B.

The sum of two subspaces is again a subspace of that space.

10. Dec 14, 2011

### csc2iffy

So, is E+B= span(EUB)=
a(2,1)+b(1,1)=(2a,a)+(b,b)=(2a+b, a+b)?

11. Dec 14, 2011