Is W a Subspace of R3? Understanding its Characteristics

[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?

 

[Office_Shredder]

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).

 

[Mark44]

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 }

 

[Mark44]

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?

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?

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.

 

[Office_Shredder]

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?

 

[Mark44]

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

 

[Mark44]

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

 

[Mark44]

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

 

[Mark44]

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.

 

[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?