- #1

- 18

- 0

## Main Question or Discussion Point

Is the basis for the subspace W of a vectorspace V spanned by the basis of the vectorspace V? If so how?

- Thread starter JamesTheBond
- Start date

- #1

- 18

- 0

Is the basis for the subspace W of a vectorspace V spanned by the basis of the vectorspace V? If so how?

- #2

radou

Homework Helper

- 3,115

- 6

What exactly do you mean? A basis for V is a span for W.Is the basis for the subspace W of a vectorspace V spanned by the basis of the vectorspace V? If so how?

- #3

matt grime

Science Advisor

Homework Helper

- 9,395

- 3

Anyway, the original question is not very clear. There is no such thing as *the* basis of a vector space. And I don't know what it means for one basis to span a basis of subset.

If you mean: given W<V, and a basis set B for V, does B have a subset that is a basis for W, then the answer is no.

- #4

- 18

- 0

I believe this is the basis extension theorem... not sure.

- #5

radou

Homework Helper

- 3,115

- 6

I'm not sure what B_W < B_V is supposed to mean, but if you meant that every linearly independent set in a vector space V (and hence a basis of a subspace W, too, since it's a linearly independent set) can be 'extended to the basis' of V, then the answer is yes.For W < V, B_W (some basis of W) is linearly independent and spans W, therefore, there exists some B_V (basis of V) s.t. B_W < B_V.

I believe this is the basis extension theorem... not sure.

- #6

- 18

- 0

Sorry, I meant to say: for some [tex] B_W \subset B_V [/tex] for some basis of V.

- #7

HallsofIvy

Science Advisor

Homework Helper

- 41,808

- 933

Let {v

- #8

mathwonk

Science Advisor

Homework Helper

- 10,820

- 985

- #9

HallsofIvy

Science Advisor

Homework Helper

- 41,808

- 933

Interesting but simple question. Since that is a one dimensional subspace and (1,0)+ (0,1)= (1,1) is in the space, {(1,0)+ (0,1)} is a basis for the subspace.

However, in case this confuses any one, the original question asked here was "if you had a basis for that subspace, how would you get a basis for R

Last edited by a moderator:

- Replies
- 3

- Views
- 2K

- Last Post

- Replies
- 4

- Views
- 5K

- Last Post

- Replies
- 5

- Views
- 2K

- Last Post

- Replies
- 8

- Views
- 1K

- Replies
- 1

- Views
- 2K

- Last Post

- Replies
- 10

- Views
- 12K

- Last Post

- Replies
- 2

- Views
- 3K

- Replies
- 2

- Views
- 3K

- Last Post

- Replies
- 7

- Views
- 5K

- Replies
- 5

- Views
- 2K