Can Spanning Subsets X and Y in Vector Space V Share No Common Vector?

In summary, if X and Y are nonempty subsets of vector space V such that span(X) = span(Y) = V, then there need not be a vector common to both X and Y. To go about proving this, one would need to find a basis for V and find a counter-example where X and Y share a vector but Span(X) = Span(Y) = {-1}.
  • #1
stunner5000pt
1,461
2
If X and Y are nonempty subsets of vector space V such taht span X = span Y = V , must there be a vector common to both X and Y? Justify.

my intuition thinks there need not be one, but i can be wrong!

to go about proving it is where the fun starts!
[tex] a_{1} x_{1} + a_{2} x_{2} + ... + a_{n} x_{n} = b_{1} y_{1} + ... + b_{n} y_{n} = V [/tex]
suppose X contaiend only one vector then ax = by. That seems to say taht one is a scalar multiple of the other
suppose X contaiend two vectors then
[tex] a_{1} x_{1} + a_{2} x_{2} = b_{1} y_{1} + b_{2} y_{2} = V [/tex]
here that means the resultant of the first addition yields the resultant of the second addition. However it doesn't mean that X = Y since ai is not bi for all i.
So there need not be any common vector

is this right? Can you help with formalizing it?
Thank you!
 
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  • #2
Maybe I got it wrong.

Let B be a basis for V. Then take X=B, Y=2B as a counter example.

Did I miss something?


*If you're not over R, replace 2 by any a!=0.
 
  • #3
Is there a fixed field you're working over, or must this work for any field, or more generally, over any ring? When the field is F = {0, 1}, then there is a vector space over F such that if Span(X) = Span(Y) = V, then Y and X share a vector, and that is when V = F. In general, if F is any field, then there is a vector space such that if X and Y are nonempty, and span(X) = Span(Y) = V, then X and Y share a vector, and that is when V = {0}. However, there are fields and vectors spaces over them such that Span(X) = Span(Y) = V, but X and Y share no common vector. To formally prove this, you just need a counter-example: V = F = R, X = {1}, Y = {-1}.
 

What is a subspace?

A subspace is a subset of a vector space that satisfies the properties of vector addition and scalar multiplication. These properties include closure, associative and commutative properties, and the existence of a zero vector and additive inverses.

How do you prove that a subset is a subspace?

To prove that a subset is a subspace, you need to show that it satisfies the properties of vector addition and scalar multiplication. This can be done by showing that the subset contains the zero vector, is closed under vector addition and scalar multiplication, and that it has additive inverses for all elements.

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

A subspace is a subset of a vector space, while a vector space is a set of vectors that satisfy certain properties. A vector space must contain a zero vector and have additive and scalar properties, while a subspace only needs to satisfy those properties within the larger vector space.

Can a subspace be infinite?

Yes, a subspace can be infinite. As long as the subset satisfies the properties of vector addition and scalar multiplication, it can be considered a subspace regardless of its size.

Are there any shortcuts to proving a subset is a subspace?

No, there are no shortcuts to proving a subset is a subspace. You must show that it satisfies all the properties of vector addition and scalar multiplication to be considered a subspace. However, there are certain properties that can help make the proof easier, such as the closure property.

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