Proving Subspace: Vectors (x,y,z) in R^3 Satisfying x+y+z=0

In summary, the conversation is about proving that a set of vectors is a subspace and finding a spanning set for that subspace. The set is (x,y,z) such that x+y+z=0 of R^3. The conversation also discusses the process of finding linear combinations to create a spanning set and clarifies that the set (0,0,0) is not necessary in the spanning set. The conversation also touches on understanding the equations and simplifying the process.
  • #1
jeffreylze
44
0

Homework Statement



Show that the following set of vectors are subspaces of R^m

The set of all vectors (x,y,z) such that x+y+z=0 of R^3 .

Then find a set that spans this subspace.

Homework Equations





The Attempt at a Solution



I managed to proof that the set of vectors is a subspace by showing that it is non-empty, closed under addition and scalar multiplication. However, I have no idea how to start on part b, how do I find a spanning set for that subspace? If I am not mistaken, I have to find linear combinations.
 
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  • #2
Ok, name one vector in the subspace. Can you find another one that's independent of the first? Can you find a third that's not a combination of those two?
 
  • #3
is (1,1,1) one of the vector? I am confused.
 
  • #4
x = -y - z
y = y
z = z

If you stare at this awhile, you might see two vectors staring back at you.
 
  • #5
jeffreylze said:
is (1,1,1) one of the vector? I am confused.
Only if 1 + 1 + 1 = 0.
 
  • #6
oh, ok. Tell me if this is right. Since x = -y-z , y=y , z=z hence (-y-z , y , z) . So x(0,0,0) + y(-1,1,0) + z(-1,0,1) , So the spanning sets are (0,0,0) , (-1,1,0) , (-1,0,1) But the given answers don't include (0,0,0) . before all that, how do you know y=y and z=z ? I only know why x = -y-z .
 
Last edited:
  • #7
Well, your set spans the subspace, but it also does so if you remove (0, 0, 0).

How did I know that y = y and z = z? The two equations are obviously true, aren't they?
 
  • #8
Ah, you got me there. They are obviously true. I was complicating stuffs, now looking back, that seemed a stupid question. OK, thanks, that did help me understand spanning sets better. Cheers
 
  • #9
If it didn't seem like a stupid question then, but now it does, I guess that means you're getting smarter, which is a good thing.
 

Related to Proving Subspace: Vectors (x,y,z) in R^3 Satisfying x+y+z=0

1. What is a subspace?

A subspace is a subset of a vector space that satisfies all the properties of a vector space, such as closure under addition and scalar multiplication.

2. How do you prove that vectors (x,y,z) in R^3 satisfying x+y+z=0 is a subspace?

To prove that a set of vectors satisfying a linear equation is a subspace, you must show that the set is closed under vector addition and scalar multiplication. This means that if you add two vectors from the set, their sum must also be in the set, and if you multiply a vector from the set by a scalar, the resulting vector must also be in the set.

3. What is the significance of x+y+z=0 in proving that a set of vectors is a subspace?

The equation x+y+z=0 represents a plane in three-dimensional space. This means that any vector in this set will lie on this plane, and any two vectors added together will result in a vector that is also on the plane. This property is essential in proving that the set is closed under addition and therefore a subspace.

4. Are there other ways to prove that a set of vectors is a subspace?

Yes, there are other ways to prove that a set of vectors is a subspace, such as showing that the set contains the zero vector, or that the set is a span of a linearly independent set of vectors. However, for this specific case of vectors satisfying x+y+z=0, proving closure under addition and scalar multiplication is the most straightforward method.

5. How is proving a set of vectors is a subspace useful in mathematics?

Proving that a set of vectors is a subspace is essential in linear algebra and other areas of mathematics. It allows us to manipulate and solve systems of equations, perform transformations, and understand the structure of vector spaces. Subspaces also have many applications in fields such as physics, computer science, and economics.

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