Can someone please refresh me on the difference between what Col A and Span A are?
Span A exists only if A is a set of vectors. It is the space of all vectors that can be written as a linear combination of vectors in A.
Col A exists only if A is a matrix- it is the space spanned by the columns of A thought of as separate vectors.
they are related, like so: an mxn matrix A can be thought of a collection of n column vectors:
A_{1},....,A_{n}, where each A_{j} is an m-vector (an mx1 column vector).
in which case, Col(A) = Span({A_{1},....,A_{n}}).
note that although a set doesn't need to be "in order", re-arranging the columns gives a different matrix. however, you will still get the same column space.
that is, if we re-arrange the columns to get A', Col(A) = Col(A') (because Span(S) is the span of a SET S, and sets don't care what order they're listed in).
it is (somewhat) important to draw a distinction between A and {A_{j}: j = 1,2,....,n}. the collection of n mx1 column vectors is a set with n things in it. the matrix A is but a single object.