Linear Algebra: Vector Space proof

In summary, to prove that the set W of all linear combinations of vectors V1, V2,...,Vk is a subspace of vector space V, we simply need to show that for any two scalars a, b and any vectors u, v in W, the linear combination au+bv is also in W. This condition can be efficiently used to prove that M is a subspace of V.
  • #1
Rocket254
33
0
Linear Algebra: Vector Space proof...

I'm really having trouble comprehending this problem. This is not exactly a "homework problem" but I need a good, formal definition of this to help with some other problems.

Let (Vectors) V1, V2,...,Vk be vectors in vector space V. Then the set W of all linear combinations of Vectors V1, V2,...Vk is a "subspace" of V.

Exactly how do you prove this?

After setting up two vectors to find the subspace, I'm lost.

Gladly appreciate any help.
 
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  • #2
Well, for some vector space V over a field F, and for the set S = {v1, ..., vk} of vectors from V, we define the span of S as the set of all linear combinations of vectors from S, i.e. span(S) = [itex]\left\{ \sum_{i=1}^n \alpha_{i} v_{i}: n \in \mathbf{N}, v_{i} \in S, \alpha_{i} \in \mathbf{F} \right\}[/itex]. Now, span(S) is obviously a subset of V, right? What simple condition must be satisfied in order for span(S) to be a subspace of V?
 
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  • #3
Do you see that if [itex]\vec{u}[/itex] and [itex]\vec{v}[/itex] are "linear combinations of V1, V2,... , Vk" then [itex]a\vec{u}+ b\vec{v}[/itex] is also a linear combination?
 
  • #4
HallsofIvy said:
Do you see that if [itex]\vec{u}[/itex] and [itex]\vec{v}[/itex] are "linear combinations of V1, V2,... , Vk" then [itex]a\vec{u}+ b\vec{v}[/itex] is also a linear combination?

Yes, I see that. I'm sorry. I am just not grasping this well at all today.

So, this would be an acceptable proof?

I'm not even exactly sure what is being asked here...

doh!
 
  • #5
Rocket254 said:
I'm not even exactly sure what is being asked here...

What's asked here is very clear, and you simply have to read definitions, that's all.
 
  • #6
Surely you've seen examples of subspaces. What do you need to prove to show that a subset of a vector space is a subspace?
 
  • #7
Ah, It just clicked.

Thank you both very much.
 
  • #8
Just making sure I have this down pat...
Could I say:

A vector V that is a subspace of "V" is a linear combination of Vectors V1,V2,...,Vk if
Vector V= C1V1 + C2V2 + C3V3 + ...+ CkVk if C1,...,Ck is all real numbers?

Would that be acceptable or do I still need to set up vectors "u and v" and write this equation for both vectors?
 
  • #9
Rocket254 said:
A vector V that is a subspace of "V" is a linear combination of Vectors V1,V2,...,Vk if Vector V= C1V1 + C2V2 + C3V3 + ...+ CkVk if C1,...,Ck is all real numbers?

What?

Look, if you have a vector space V and some *subset* M of V, then this very subset M is a *subspace* of V if and only if, for every two scalars a, b and vectors u, v from M, au+bv is in M, too. (One can easily show that all the other vector space axioms follow from this condition, which makes this condition efficient on an operative level.)
 

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of objects called vectors, which can be added together and multiplied by numbers called scalars. Vector spaces are used to model geometric concepts and operations in mathematics, physics, and engineering.

2. What are the axioms of a vector space?

The axioms of a vector space are a set of rules that must be satisfied for a set to be considered a vector space. These axioms include closure under vector addition and scalar multiplication, associativity and commutativity of addition, existence of an additive identity element, and existence of a multiplicative identity element for scalars.

3. How do you prove that a set is a vector space?

To prove that a set is a vector space, you must show that it satisfies all of the axioms of a vector space. This can be done by checking each axiom individually and showing that it holds for all elements in the set.

4. What is the difference between a subspace and a vector space?

A subspace is a subset of a vector space that also satisfies the axioms of a vector space. This means that all the operations defined in the vector space can also be performed on the elements in the subspace. A vector space, on the other hand, is a set that satisfies all the axioms of a vector space, but may not necessarily be a subset of another vector space.

5. How is linear independence related to vector spaces?

Linear independence is a property of a set of vectors in a vector space. It means that none of the vectors in the set can be written as a linear combination of the others. This property is important in vector spaces because it allows us to determine the number of basis vectors needed to span the entire space, and also helps in solving systems of linear equations.

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