# Proof of subspaces

1. Nov 13, 2008

### shutoutsteve

a) {(x1,x2,x3) | x1+x2 ≥ 0}

b) {x∈R3 |proj(1,1,1) (x) ∈ Sp({(1,1,1)})}

Prove the set is or is not a subset of R n

I have no idea how to solve this. Our textbook gives NO examples of how to prove these

Thanks- Steve

2. Nov 13, 2008

### Staff: Mentor

Are you sure this is what you have to prove?
Does your book have any definitions that are related to these problems?

3. Nov 14, 2008

### shutoutsteve

That is what my book says exactly. :(

4. Nov 14, 2008

### HallsofIvy

Staff Emeritus
Do you understand that you have just said that your book contains that one problem and a lot of blank pages? I don't believe for a moment that your book does not have a definition of "subspace". Try looking in the index!

5. Nov 14, 2008

### Staff: Mentor

I don't believe that your book says "Prove the set is or is not a subset of R n"

6. Nov 15, 2008

### shutoutsteve

...oops...
...
Our book does have a definition, but it is very vague, about a paragraph long and then it goes on about something else. There aren't even any examples that LOOK like this in our book. It's ok though because we are covering this Monday in class, and 14/65 people handed in the assignment with that question because nobody understood it.

7. Nov 16, 2008

### Staff: Mentor

Look at the definition.
Definitions in math books are almost never "very vague". On the contrary, they are generally very precise. For many types of problems, if you understand the definition, you don't really need an example.

If you take a careful look at the definition of a subspace of a vector space, and make an honest effort at this problem I'm sure you'll get some help from us.

8. Nov 16, 2008

### HallsofIvy

Staff Emeritus
A subset, U, of a vector space V, is a subspace if and only if
1) It is non-empty. (Equivalently, it contains the 0 vector)
2) It is closed under addition. (If x and y are in the set so is x+ y)
3) It is closed under scalaar multiplication. (If x is in the set so is ax for any a in the underlying field [here, the real numbers])

I'll be that's essentially what your book has.

Which of the sets given in a and b above satisfy those?

9. Nov 17, 2008

### shutoutsteve

That is the definition we received today in class, which was similar to what the textbook said, except it wasn't laid out as neat and orderly as this one. I now understand the topic.

I apologize for being stubborn earlier about the textbook quotes. I realize now all i needed was this definition, one that I could understand given the material covered so far in class.
Thanks