# Why is W a subspace of R3?

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?

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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).

Mentor
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 |
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?

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?
Note that your matrix is 3 X 1. IOW, it has only a single column.
Or would one need to solve this system using Gaussian Elimination to show whether or not W is a subspace?

Have you given us a complete description of the problem?

jreis
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).
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

Note that your matrix is 3 X 1. IOW, it has only a single column.
This would be a 3x3 matrix. I don't see why you're confused?

Have you given us a complete description of the problem?
Yes. W = { (3 given equations) : x, y, z are arbitrary constants }

Mentor
Note that your matrix is 3 X 1. IOW, it has only a single column.
jreis said:
This would be a 3x3 matrix. I don't see why you're confused?
Because x+2y+3z, 4x+5y+6z, and 7x+8y+9z each represent a single number.

Have you given us a complete description of the problem?
jreis said:
Yes. W = { (3 given equations) : x, y, z are arbitrary constants }
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?

jreis
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?
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.

Staff Emeritus
Gold Member
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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?

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?

jreis
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?

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?

jreis
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?

yeah

Mentor
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?

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?

jreis
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?

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

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.

jreis
$$c_1\begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix} + c_3 \begin{bmatrix} 3 \\ 6 \\ 9\end{bmatrix}$$
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.

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

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.