I have a question I need help with. How can I show the following?
Why isAx = b solvable exactly when the column spaces of A and
[A b]are equal?
Well, what does it mean for a vector b to be "in the column space" of matrix A? - It means (by definition of "column space of A"), that that vector b is a linear-combination of the column vectors of matrix A. This is the case if and only if the addition of b to the column vectors of A does not generate a larger vector space (that, of course, contains b).
Further, saying that the column spaces of A and [A b] (the matrix A with b appended as a column) are the same simply means that b was already in the column space of A. The vector space spanned by the columns of A thought of as vectors, the "column space of A" is the set of all possible vectors of the form Av for all v. You can see that by considering [itex]Ae_i[/itex] for [itex]e_i[/itex] a member of the "standard basis", the vector with "1" in the ith place, "0" elsewhere.