1. ## Question Column-spaces

I have a question I need help with. How can I show the following?

Why is
Ax = b solvable exactly when the column spaces of A and

[A b]
are equal?

2. Originally Posted by mathfilip
I have a question I need help with. How can I show the following?

Why is
Ax = 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).

3. 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 $Ae_i$ for $e_i$ a member of the "standard basis", the vector with "1" in the ith place, "0" elsewhere.