Are These Linear Algebra Statements True or False?

[ammar555]

I have these questions on my studyguide and I know both of them are false. I just don't have a good counterexample or a good explaintion to prove so.

1) two subsets of a vector space V that span the same subspace of V are equal.

False: They don't have to be equal

2) The union of any 2 subspaces of a vector space V is a subspace of V.

False: Adding two subspaces doesn't necessary mean they will stay inside the vector space

 

[Alesak]

1) +1 and -1 in R
2) x and y axes in R^2. Addition and set union of two vector subspaces are not equal notions, though.

 

[ammar555]

1) +1 and -1 in R
2) x and y axes in R^2. Addition and set union of two vector subspaces are not equal notions, though.

Please I need a little more explaintions, I don't fully understand your examples. And does that mean my answer to number 2 is wrong?

Thanks though!

 

[Alesak]

1) real line R is a vector space, and it is generated by both +1 and -1. So these two one-element subsets {1} and {-1} generate R yet are different.

2) it's correct, just unclear. Consider, for example, subspaces of R^2 that coincide with usual x and y axes. Vector (1, 0) is on x axis, vector (0, 1) on y axis, but if you add them they lie out of both subspaces.

So set theoretic union of two vector subspaces doesn't have to be vector subspace. Perhaps you can try to work out operation that makes some kind of "union" of two vector subspaces again a subspace.

You can surely find more information about vector spaces on wikipedia or in your textbook.

 

[ammar555]

1) real line R is a vector space, and it is generated by both +1 and -1. So these two one-element subsets {1} and {-1} generate R yet are different.

2) it's correct, just unclear. Consider, for example, subspaces of R^2 that coincide with usual x and y axes. Vector (1, 0) is on x axis, vector (0, 1) on y axis, but if you add them they lie out of both subspaces.

So set theoretic union of two vector subspaces doesn't have to be vector subspace. Perhaps you can try to work out operation that makes some kind of "union" of two vector subspaces again a subspace.

You can surely find more information about vector spaces on wikipedia or in your textbook.

Much better explanations precisely for number 1, I think I understand it now. Number 2 I'm still shady about it. Maybe if I read more about vectors, I can then understand it more clearly.

Thanks a million times!

 

[Alesak]

Much better explanations precisely for number 1, I think I understand it now. Number 2 I'm still shady about it. Maybe if I read more about vectors, I can then understand it more clearly.

Thanks a million times!

Sure, get any good textbook on linear algebra, it will be crystal clear to you in no time.