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

[stanford1]

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.

 

[Mark44]

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.

 

[stanford1]

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.

 

[Mark44]

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.

 

[HallsofIvy]

{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.

 

[HallsofIvy]

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.