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?
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.