Suppose that you know two variables x and y have linear relationship that is y= mx +c . Suppose in an experiment you obtained the following data:
If you want to fit a line with equation y= mx +b through these points. As you learned in Laboratary 5, you need to form the following system of linear equations.
This is an overdetermined system. That is it has more equation than unknowns. Over determined systems usually are inconsistent.
Writing the matrix equation for this linear system we get
Find the reduced row echelon form of the augmented matrix.
As you see the rank of rref of the augmented matrix is 3 while the rank of the coefficient matrix is 2. Therefore the system is inconsistent. So there is no x such that Ax=b or Ax - b =0. Now the question is : Since Ax - b is non-zero we try to make it as small as possible. So the goal is to find an approximation, Ax, for b or to minimize Ax-b?
Restating the question, how could we find an x such that Ax is as close as possible to b?