Do Equal Dimension Subspaces Imply Equality?

[pivoxa15]

A and B are two subspaces contained in a finite vector space V and

dimA = dimB

Can we conclude A=B?

In that subspaces A and B are really the same subspace and every element in one is in the other?

I think yes because if dimA=dimB then their basis will contain the same number of vectors. These vectors span their respective subspaces. Given that each basis set contain an equal number of linearly indepedent vectors, they can be reduced to the standard vectors in V. Hence both spaces can be spanned by a single set of standard vectors in V so space A=B.

 

[HallsofIvy]

If V has dimension greater than 1, then it has at least 2 independent vectors- that is it has a basis {v₁, v₂,...}.

Let A be the subspace spanned by v₁ and let B be the subspace spanned by v₂ so that they both have the same dimension, 1. Are A and B equal? That is, do they contain exactly the same vectors?

Here's your reasoning error: " Given that each basis set contain an equal number of linearly indepedent vectors, they can be reduced to the standard vectors in V."
That's not true. In the xy-plane, the x-axis is the subspace spanned by i and y is the subspace spanned by j. Yes, they can be reduced to the "standard" vectors in V but not the same standard vectors. Even more, the line y= x is a subspace, spanned by the single vector i+ j. It's basis cannot be reduced to a subset of the standard vectors.

 

[pivoxa15]

I see what you are getting at. What about if I add that A is a subspace of B. And if dimA = dimB than A=B.

 

[HallsofIvy]

Yes, if A is a subspace of B and has the same dimension, then A= B.