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Why is W a subspace of R3?

  • Thread starter jreis
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  • #1
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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?
 

Answers and Replies

  • #2
Office_Shredder
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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).
 
  • #3
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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 |
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?
 
  • #4
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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).
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 }
 
  • #5
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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?
 
  • #6
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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
Office_Shredder
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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?
 
  • #8
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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
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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
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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?
yeah
 
  • #11
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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
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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
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$$c_1\begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix} + c_3 \begin{bmatrix} 3 \\ 6 \\ 9\end{bmatrix}$$
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
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$$c_1\begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix} + c_3 \begin{bmatrix} 3 \\ 6 \\ 9\end{bmatrix}$$
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
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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
HallsofIvy
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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?
 

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