So, a subspace is a set of vectors that satisfies:

1) It contained the zero vector;

2) It's closed under addition and subtraction.

By "closed", it means that when I add another vector in R

^{2}or multiply by a scalar k on A(x)=m, it will end up with A(x)+A(y)=m or k(A(x))=m.

Is it correct so far?

Now, I know span is like mapping, but I don't have a precise definition nor explanation for it...Is it like mapping a spanning set to a set of column spaces?

Range is all possible result of a linear transformation of a matrix, ie. L(x)=b, where b is a spanned vectors..is it?

I need some clear definitions and examples to sort this out...Thanks! :D