Dimension of a variety

This should be a simple question for anyone familiar with basic algebraic geometry, but the concept is getting the best of me. Basically I am unclear on the concept of dimension of a variety. I'm still stuck in the algebraic mindset that dimension is the (minimum) number of elements needed to span the space, but that doesn't seem to carry over. I have several books on the subject, and the most intuitive definition I've found was that the dimension of a variety is the smallest r such that the variety is isomorphic to projective r-space. However, this still has me confused: how does one compute/find/understand the dimension of a projective space? I know projective n space has a defined dimension, but I can't understand why...

In particular, one knows a smooth point of a variety is a point at which the dimension of the (affine) tangent space is locally constant. I can translate this back into the original vector space and see the dimension, but I don't see how this carries over into projective space.

I suppose I'm looking for a little intuition in this matter...

Also, I'm posting this in the "Linear and Abstract Algebra" forum also. Mods, if this is contrary to any rules, please remove whichever posting is in the wrong place, but since this is an algebraic geometry question, I figured it would be best to get both perspectives.

Thanks in advance.

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