> I don't understand the + in the first reply!

If X and Y are vector spaces (e.g., a line or a plane through the origin), then X+Y is the sum of all elements of each.

If you think about this example (dim(X)=1 and dim(Y)=2) you can easily recover the first formula intutively. There are two cases for the formula

   dim(X+Y) = dim(X) + dim(Y) - dim(X ∩ Y)

Case 1: X is a line inside the plane Y. In that case X+Y=Y and X∩Y=X, and the formula of the dimensions becomes 2=1+2-1

Case 2: X intersects Y at the origin. In that case X+Y is a three-dimensional space and X∩Y is 0, thus the formula says 3=1+2-0.

And you need some rule for what summing the elements means, where does it come from, I'm curious? If it's just a formal sum then the elements are incomparable between different spaces - and we can't say that one contains the other.

I think I must understand when you say "X + Y" is the sum of elements that you're not using a formal sum {x} + {y} of each element x, and y but you mean sort of like the union of elements in X and Y?

No, the idea is that they are all vectors in the same all-encompassing vector space. You can always sum them. What's missing on the formula is that all of these vector spaces are subspaces of the same larger one. That's implicit, otherwise the sum wouldn't make sense. Think that you are playing inside R^n and juggling a few lower-dimensional subspaces of it.

You can think of each vector space as being embedded in the direct sum of the two.

That makes sense, thanks!

> I don't understand the +

To add to the confusion, in some contexts the sum is the same as ("isomorphic to") the product!