U+v in subspace W, is u or v in subspace

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Homework Help Overview

The discussion revolves around the properties of subspaces in vector spaces, specifically whether the sum of two vectors being in a subspace implies that at least one of the vectors is also in that subspace. Participants explore this concept in the context of linear algebra.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants question the implications of the sum of two vectors being in a subspace and whether this necessitates that either vector must also belong to the subspace. Counterexamples are sought to illustrate the point.

Discussion Status

Some participants have offered clarifications regarding the nature of vectors and their sums in relation to subspaces. There is an ongoing exploration of examples and counterexamples to better understand the properties of subspaces.

Contextual Notes

There is mention of specific subspaces, such as the zero vector space and subsets of R², which may influence the discussion. Participants also note the terminology used when discussing closure properties of vector spaces.

stanford1
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1. Homework Statement

My question is if u+v is in the subspace can you say that u or v is in the subspace? If not would there be a counterexample? 2. Homework Equations

closed under addition/scalar multiplication

3. The Attempt at a Solution

I know that if u or v were in the subspace they would be closed under addition or multiplication. I don't know if you can say the same for (u+v) and apply it just to u or v.

Thank you for any help.
 
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stanford1 said:
1. Homework Statement

My question is if u+v is in the subspace can you say that u or v is in the subspace? If not would there be a counterexample?


2. Homework Equations

closed under addition/scalar multiplication

3. The Attempt at a Solution

I know that if u or v were in the subspace they would be closed under addition or multiplication. I don't know if you can say the same for (u+v) and apply it just to u or v.

Thank you for any help.

{0} is a subspace of R, a one-dimensional vector space. Are there vectors in R, that add to 0, that aren't in the subspace?

BTW, we don't talk about vectors being closed under addition or scalar multiplication - we talk about the space they belong to as being closed under addition or scalar multiplication.
 
Just making sure I have this correctly, that would mean that a or b is not in the vector space, just a+b. Thank you for the quick response.
 
Don't think of a + b as being two things: it's a single thing. a and b are two vectors that happen to add up to whatever value a + b represents.
 
Mark44 said:
{0} is a subspace of R, a one-dimensional vector space. Are there vectors in R, that add to 0, that aren't in the subspace?
No, its as 0 dimension vector space. But your point is correct.

BTW, we don't talk about vectors being closed under addition or scalar multiplication - we talk about the space they belong to as being closed under addition or scalar multiplication.
 
For example, the subset of R2, {(x, y)|y= x} is a subspace. The vectors (1, 0) and (1, 2) are not in that subspace but their sum, (2, 2), is.
 

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