Proving W=V When W is a Subspace of an n-Dimensional Vector Space

[mlarson9000]

Homework Statement

Prove that if W is a subspace of an n-dimensional vector space V and dim(W) = n, then W=V

Homework Equations

The Attempt at a Solution

I don't know where to start.

 

[Dick]

What does 'dim' mean, and what does subspace mean? Can you look it up? That should get you started.

 

[mlarson9000]

I understand why W would equal V. Every linear combination of vectors in W is in W, and since W and V have the same number of elements in a basis, and W is in V, then W=V. I just don't know how to illustrate that.

 

[g_edgar]

Probably the text, just before this, has some results about "linear independence", "spanning sets", and "dimension". Probably there is a proof that shows any two bases for a given vector space are the same size. That material is what you need to understand in order to do this problem.

 

[HallsofIvy]

As g edgar suggested, and I will make even more specific:

If any two of these are true, so is the third
1) There exist a set of n independent vectors
2) There exist a set of n vectors that span the space
3) The dimension is n