Understanding Subspace Basis and Counterexample

[Caspian]

My book made the following claim... but I don't understand why it's true:

If v_1, v_2, v_3, v_4 is a basis for the vector space \mathbb{R}^4, and if W is a subspace, then there exists a W which has a basis which is not some subset of the v's.

The book provided a proof by counterexample: Let v_1 = (1, 0, 0, 0) ... v_2 = (0, 0, 0, 1). If W is the line through (1, 2, 3, 4), then none of the v's are in W.

Is it just me, or does this not make any sense? First of all, (1,2,3,4) is a linear combination of (1,0,0,0)...(0,0,0,1), isn't it?

I'm very confused...

Any help would be greatly appreciated :).

 

[Fredrik]

Yes, but that's not what they're saying. They're saying that none of the four v_i vectors are in W. They're also saying that a basis vector of W (which must be a multiple of (1,2,3,4)) can't be equal to one of the v_i.

When they talk about the set of "v's" they really mean a set that only has four members, not the subspace they span.

 

[Caspian]

Ok, ok... I get it. (1,2,3,4) is in the vector space spanned by (1,0,0,0)...(0,0,0,1), but these vectors aren't a basis for the subspace which is the line through (1,2,3,4) because these vectors span too much space.

Ok, I guess this was a dumb question. Thanks for your help :).

 

[Landau]

I don't think you fully get it yet.

Ok, ok... I get it. (1,2,3,4) is in the vector space spanned by (1,0,0,0)...(0,0,0,1), but these vectors aren't a basis for the subspace which is the line through (1,2,3,4) because these vectors span too much space.

It should be obvious that B={(1,0,0,0),...,(0,0,0,1)} isn't a basis for W, since B spans the whole space R^4! So they can't be a basis for any proper subspace of V (indeed, they span "too much space").

But that's not what your book is asserting. They are only talking about a subset of B={(1,0,0,0),...,(0,0,0,1)}. So they're saying that even some subset of B cannot be a basis of W. Remember that a basis of W first of all consists of elements of W. None of the vectors in B are in W.

 

[HallsofIvy]

The point you need to keep in mind is that there exist an infinite number of different bases for any given vector space. If W is a subspace of V and we are given a basis for W, then we can extend that to a basis for V. That is, the basis for V will consist of all vectors in the basis for W together with some other vectors. But there can also exist bases for V that do not contain any of the vectors in the basis for W.