Homework Help: Why is W a subspace of R3?

1. Oct 15, 2013

jreis

I want to know why this subset W is a subspace of R3.

W is defined as:

| x+2y+3z |
| 4x+5y+6z |
| 7x+8y+9z |

I know the possible subspaces of R3 are the origin itself, lines through the origin, and planes through the origin. Would W be a subspace of R3 simply because there would be no coefficient column to this matrix? Or would one need to solve this system using Gaussian Elimination to show whether or not W is a subspace?

2. Oct 15, 2013

Office_Shredder

Staff Emeritus
Do you know what the definition of a subspace is? You don't need to prove this is a line or plane through the origin, you just need to show that it fits the general definition of a subspace (which is typically way easier).

3. Oct 15, 2013

Staff: Mentor

This doesn't make sense to me. A set is normally defined by some rule that indicates what things are in the set, and what things aren't. How can I tell if some vector <x, y, z> is in set W or not?

Note that your matrix is 3 X 1. IOW, it has only a single column.
Have you given us a complete description of the problem?

4. Oct 15, 2013

jreis

Hmm, I think for W to be a subspace it needs to:
1) Contain the 0 vector
2) for vectors v and w in W, v+w is also in W
3) for vector v in W, and any real constant c, cv is also in W

This would be a 3x3 matrix. I don't see why you're confused?

Yes. W = { (3 given equations) : x, y, z are arbitrary constants }

5. Oct 15, 2013

Staff: Mentor

Because x+2y+3z, 4x+5y+6z, and 7x+8y+9z each represent a single number.

There is not a single equation in your first post. For example, x + 2y + 3z is NOT an equation. An equation has an = symbol in between two expressions.

So what is the exact problem description?

6. Oct 15, 2013

jreis

Expressions then.. This is an exact problem out of my textbook, and I'm trying to figure out why W is a subspace holds true.

7. Oct 15, 2013

Office_Shredder

Staff Emeritus
jreis, is W the set of vectors of the form

| x+2y+3z |
| 4x+5y+6z |
| 7x+8y+9z |

where x,y and z are arbitrary real numbers or something?

8. Oct 15, 2013

Staff: Mentor

Your set W is the set of all linear combinations or these three vectors:
$$\begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix}, \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix}, \begin{bmatrix} 3 \\ 6 \\ 9\end{bmatrix}$$

Is the zero vector in this set?
If you take two arbitrary vectors in the set, is their sum also in this set?
If c is any scalar, and v is in the set, is cv also in the set?

9. Oct 15, 2013

jreis

This is what I'm asking. How do I approach these questions..? I know the 0 vector is in the set, but the other two?

10. Oct 15, 2013

jreis

yeah

11. Oct 15, 2013

Staff: Mentor

For any two arbitrary vectors in the set, show that their sum is also in this set.
If c is any scalar, and v is an arbitrary vector in the set, show that cv also in the set.

Do you know how to write an arbitrary vector in set W? Is that what's blocking you?

12. Oct 15, 2013

jreis

Yes. I guess that's what I'm confused about

13. Oct 15, 2013

Staff: Mentor

$$c_1\begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix} + c_3 \begin{bmatrix} 3 \\ 6 \\ 9\end{bmatrix}$$
is an arbitrary vector in W.

You can also write it like this: c1<1, 4, 7> + c2<2, 5, 8> + c3<3, 6, 9>, where it's understood that the vectors are column vectors.

14. Oct 15, 2013

jreis

Right, I get that. But if I were to add a second vector, how could I show that their sum is also in W?

15. Oct 15, 2013

Staff: Mentor

The same way you can tell whether any vector is in W. A vector is in W if it is a linear combination of the three vectors I show in post #13.

16. Oct 16, 2013

HallsofIvy

If u is some vector in this set then u= a<1, 4, 7>+ b<2, 5, 8>+ c<3, 6, 9> for some numbers a, b, c. If v is a vector in this set then v= d<1, 4, 7>+ e<2, 5, 8>+ f<3, 6, 9> for some numbers d, e, f.

So what is u+ v? How do you know it is in the same set?