Basis of Vector Space V: Subset of B = Basis of U?

[e(ho0n3]

Let B be a basis of a vector space V. If U is a subspace of V, is it true that a subset of B may serve as a basis for U?

 

[LukeD]

Let V be R^2. One basis of R^2 is {(1,0),(0,1)} (we'll call this B). Let U be {(x,y): y = 2x}. This is a subspace of R^2. One basis of U is {(1,2)}, and every basis of this space contains a multiple of the vector (1,2). Since no multiple of (1,2) is in B, there is no subset of B that is a basis of U.

However, it is possible to have a subset of B be a basis for U. For instance, if B were {(1,2),(2,1)} (this is indeed a basis of V), then {(1,2)} is a basis for U and a subset of B.

 

[n_bourbaki]

Let B be a basis of a vector space V. If U is a subspace of V, is it true that a subset of B may serve as a basis for U?

This question is so trivially false that it tells me that you didn't bother to think about any concrete examples at all. The first thing you do if you're not sure about something is you see what happens in a case you can work out by hand, if at all possible. A trivial consequence of this statement would be that a vector space only had a finite number of subspaces, for heaven's sake. I apologise for sounding harsh.

 

[tiny-tim]

Let B be a basis of a vector space V. If U is a subspace of V, is it true that a subset of B may serve as a basis for U?

Hi e(ho0n3!

Hint: take a really easy example to visualise: let U be ordinary space (R³), and let B be (1,0,0) (0,1,0) and (0,0,1).

Can you draw a two-dimensional subspace which doesn't include any of B?

 

[HallsofIvy]

Be careful exactly what you are saying. If U is a proper subspace of V, then a basis for V cannot be a basis for U because it will span a space, V, larger than U.

What is true is that there always exists a basis for V that contains a basis for U.

 

[tiny-tim]

What is true is that there always exists a basis for V that contains a basis for U.

Hi HallsofIvy!

A basis, yes, but not necessarily the given basis.

 

[e(ho0n3]

I have a hard time coming up with examples or counterexamples. Thanks guys.

 

[n_bourbaki]

Does that mean you didn't understand LukeD's example? Or you don't understand that in a vector space of dimension 2 with basis {u,v} then not evey subspace is spanned by either u or v, which is surely the most obvious and trivial example you should have first thought of. Or you do, but just didn't think of them yourself and you're explaining why?

Surely it is clear that if I have n (linearly independent) vectors, then subsets of these span exactly 2^n possible vector subspaces? And that almost all vector spaces have a lot more subspaces than that?

 

[e(ho0n3]

Or you do, but just didn't think of them yourself and you're explaining why?

This one.

Surely it is clear that if I have n (linearly independent) vectors, then subsets of these span exactly 2^n possible vector subspaces? And that almost all vector spaces have a lot more subspaces than that?

Are you asking? The first one is clear to me. The second one isn't.

 

[n_bourbaki]

In one thread you're asking about how to distribute operators over tensor products in relation to quantum computing, and in another you're do not know that a vector space of dimension at least 2 (over something like the field of complex numbers) has infinitely many distinct subspaces? This bothers me.

Consider R^2. How many lines through the origin are there?

 

[e(ho0n3]

In one thread you're asking about how to distribute operators over tensor products in relation to quantum computing, and in another you're do not know that a vector space of dimension at least 2 (over something like the field of complex numbers) has infinitely many distinct subspaces? This bothers me.

It bothers me a lot more.

Consider R^2. How many lines through the origin are there?

Countless. I understand now. Thus, for any n-dimensional space V, since it is isomorphic to Rⁿ, it contains a subspace isomorphic to R², and since R² contains infinitely many subspaces, V has infinitely many subspaces. Right?

 

[n_bourbaki]

Thus, for any n-dimensional space V, since it is isomorphic to Rⁿ,

any n-dimensional real vector space. But a similar analysis for any field will tell you something. But the 2 wasn't important: R^n contains infinitely many proper subspaces of any dimension.