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Expert: - 11/12/2008


Question
If V is a finite dimensional vector space over F and if W is a subspace of V, prove that:

a)   W is finite dimensional over F and dimF (W) <= dimF (V).

b)   If dimF (W) = dimF (V) , then V = W.

I can logically reason why this is so but feel less than in writing a formal proof. If W is a subspace of V then W is a subset of V and whatever dimension V has W has to be the same or less since W is contained in V. If W is V then the dim W = dim V but this is not what I am to prove I am to prove if dim V = dim W then V = W. Can you help me Scott?

Thanks,
Sombra

Answer
a) Since V is finite dimensional and W is a subspace of V, then W is also finite dimensional.  Since W is a subspace of V, the dimF(W) must be less than or equal to dimF(V).  If it was not, then there would be a subset of W that was not in V.

I will include the following definitions for b:

1) basis for a subspace:
A basis for a subspace W is a set of vectors {v1, ...,vk}
in W such that: {v1, ..., vk} is linearly independent;
and {v1, ..., vk} spans W.

2) dimension of a subspace:
The dimension of a subspace W is the number of vectors in any basis of W. (If W is the subspace {0}, we say that its dimension is 0.)

3) subspace:
A subset W of Rn is a subspace of Rn if:

 the zero vector is in W;
 x+y is in W whenever x and y are in W; and
 ax is in W whenever x is in W and a is any scalar.


b) Since V is a finite dimensional subspace and
W is a subspace of V.
Lets say that the dimF(V) is n.  This means that there are n vectors such that every element in V can be expressed as the summation
a1*v1 + a2*v2 + ... + an*vn.  The also applies to W, since the dimF(W) is also n.  Since W is a subset of V, then V must also be a subset of W since we have n distinct vectors.  Since they are subsets of each other, they are indeed the same vector space.

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