Is W a Subspace of R3? Understanding its Characteristics

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In summary, the conversation discusses the determination of whether the subset W, defined as a set of vectors in R3, is a subspace. The possible subspaces of R3 are mentioned, and it is questioned whether W fits the general definition of a subspace. The discussion then focuses on the specific definition of W and how to approach proving that it is a subspace by showing that it contains the zero vector and that it is closed under addition and scalar multiplication. The conversation ends with a clarification on how to write an arbitrary vector in W.
  • #1
jreis
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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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  • #2
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
jreis said:
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?


jreis said:
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.
jreis said:
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?
 
  • #4
Office_Shredder said:
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


Mark44 said:
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?

Mark44 said:
Have you given us a complete description of the problem?
Yes. W = { (3 given equations) : x, y, z are arbitrary constants }
 
  • #5
Mark44 said:
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.

Mark44 said:
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?
 
  • #6
Mark44 said:
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.
 
  • #7
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
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
Mark44 said:
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?
 
  • #10
Office_Shredder said:
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
 
  • #11
jreis said:
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?
 
  • #12
Mark44 said:
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
 
  • #13
$$c_1\begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix} + c_3 \begin{bmatrix} 3 \\ 6 \\ 9\end{bmatrix}$$
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
Mark44 said:
$$c_1\begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix} + c_3 \begin{bmatrix} 3 \\ 6 \\ 9\end{bmatrix}$$
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?
 
  • #15
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
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?
 

1. What is a subspace?

A subspace is a subset of a vector space that also satisfies the properties of a vector space. In other words, a subspace is a smaller space that is contained within a larger vector space and follows the same rules and operations as the larger space.

2. How do you determine if W is a subspace of R3?

To determine if W is a subspace of R3, we must check if it satisfies the three properties of a subspace: closure under addition, closure under scalar multiplication, and containing the zero vector. This can be done by checking if all vectors in W can be added and multiplied by scalars to get another vector in W, and if the zero vector is also in W.

3. Why is it important to know if W is a subspace of R3?

Knowing if W is a subspace of R3 is important because it allows us to use the properties and operations of vector spaces in W. This can help us analyze and solve problems that involve vector operations in a more efficient and organized manner.

4. Can W be a subspace of R3 if it is not a subset of R3?

No, W must be a subset of R3 in order to be a subspace. This is because all vectors in W must also be elements of R3 in order to satisfy the properties of a vector space, such as closure under addition and scalar multiplication.

5. How can we prove that W is a subspace of R3?

We can prove that W is a subspace of R3 by showing that it satisfies the three properties of a subspace: closure under addition, closure under scalar multiplication, and containing the zero vector. This can be done by using mathematical proofs or by showing specific examples of how the properties are satisfied in W.

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