Vector Space Subspace Basis: Finding Compatible Bases

[iamalexalright]

Homework Statement

Let S be a subspace of a vector space V. Let B be a basis for V. Is there a basis C for S such that $$C \subseteq B$$?

not really sure how to approach this... any hints?

 

[Mark44]

Homework Statement

Let S be a subspace of a vector space V. Let B be a basis for V. Is there a basis C for S such that $$C \subseteq B$$?

not really sure how to approach this... any hints?

If there is another basis C for V, what must be true about the two bases, B and C?

 

[iamalexalright]

They are isomorphic to each other... have the same cardinality...

Dunno where you are going with that

 

[Mark44]

Cardinality is where I'm going. Could there be a basis C with fewer members than B has? Could there be a basis C that is a proper subset of B?

 

[iamalexalright]

hrm - i'll use an example to make it a little more clear for me

well if my vector space is $$R^{3x1}$$ and let S be a subset of this space in which the third entry in the vectors is zero.

So a basis (call it B) of the vector space is simply the standard basis (e1 = (1,0,0), e2 = (0,1,0), e3 = (0,0,1)).

A basis(call it C) for the subspace is simply e1 and e2.

So C has less members AND C is a proper subset of B.

 

[Mark44]

The vector space V in your example is really R³, and S = {(x₁, x₂, 0}}.

Sure, B = {<1,0, 0>, <0, 1, 0>, <0, 0, 1>} is a basis for V, and C = {<1, 0, 0>, <0, 1, 0>} is a basis for S. So for your example $$C \subseteq B.$$

Somehow I misread your first post to mean that B and C were both bases for S.

 

[iamalexalright]

Is this true in general? Or is it only true case by case?

Another similar question:
Given a basis a subspace S of a vector space V. If C is a basis for S can I, in general, add vectors to the basis C to get a basis B for V? (and I'm talking about finite vector spaces)

Seems like it should be true but I can't give a formal proof.

 

[Mark44]

Yes, unless S happens to be the vector space itself. Think about it in terms of some simple, easy to visualize spaces, with V = R³ and S a subspace of R³ spanned by some plane through the origin. If u1 and u2 make up a basis for S, then adding a vector not in the plane (not in Span(u1, u2)) gets you a basis for the entire space V.

 

[HallsofIvy]

If S is a subpace of V and B is a basis for V, then it is NOT necessarily true that there exist a subset of B which is a basis for S. For example, {(1, 0), (0, 1)} is a basis for $R^2$. If S= {(x, y)| x= y}, it is a (one dimensional) subspace of $R^2$ but neither (1, 0) not (0, 1) is a basis vector for it.

The other way, "if S is a subspace of V and B is a basis for S, then there exist a basis for V containing B", is true.