Can someone help with some linear algebra theory?

I am a computer science student at San Francisco State. This means I have to take a MILLION math classes.
I am on my second try at linear algebra and I thought I’d better ask anyone and everyone as many questions as I can.

We’re getting close to the half way point in the class and I believe I am suffering from not fully understanding the concept of Rn. I understand R2 is a two dimensional plane and R3 is a three dimensional plane etc., but is this different from a matrix with dimension 1 or 2?

I am a little fuzzy about transformations and how a subspace of R3 can be composed of vectors that ‘live’ in R2…

These are my preliminary questions and im looking for those helpful hints and tricks to help someone understand because the theorems for a lot of this stuff make sense, but at times they are too abstract for me to visualize for that ‘complete’ understanding.

Think of my understanding thus far as a slice of swiss cheese; it definitely exists, but full of holes :)