Question

\(\bf Problem\ 10.1\) (PDF from online course): (3.6 #11. Introduction to Linear Algebra: Strang) A is an m by n matrix of rank r. Suppose there are right sides b for which Ax = b has no solution.
a) What are all the inequalities (< or ≤) that must be true between m, n, and r?
Solution:
a) The rank of a matrix is always less than or equal to the number of rows and columns, so r ≤ m and r ≤ n. The second statement tells us that the column space is not all of Rn, so r < m.
\(\bf{Question:}\) What does it mean that there is a right-side b for which Ax=b has no solutions? Is it an arbitrary b among the b vectors that is not equal to Ax? If yes, why would such a non-solution b be included in the first place?

Answer 1

It is a vector b that you cannot achieve via any linear combination of the column vectors of A. Equivalently, b is not in the column space of A.
Example:
[1 0] =1
[0 0] =1
No linear combination c*(1,0)+d*(0,0) will produce b=(1,1)

Answer 2

To answer your second question, such a 'non-solution' would just be a fact of life: you perform some experiment and get a bunch of data points (x,y) that don't necessarily agree i.e. they're not collinear. So, Ax=b has no solution, but the projection of b onto the column space of A will have a solution, with some error of course (this is in section 4.3 - least squares) of the book. I hope that helps.

Answer 3

I think ovanbc's replies are correct, but maybe another way of saying it might be helpful? I think the problem could have been written: "Suppose there exist vectors b for which Ax = b has no solution." The point is that the column space of A does not contain every possible m x 1 vector. Since that's the case, we know r < m. Hope that helps.

Answer 4

Belated reply . . . . both commentators supply good points. Thanks.