Least Squares Solutions
You see that so the singular values are and .
Since , can't send anything in the direction of .
This tells you that sends every vector to a multiple of .
You can see that .
Since isn't a multiple of you know that has no solution.
Use the graphic to find an that makes as close as possible to .
Since sends every to a multiple of , the closest that can come to is . This is the projection of on the column space of and is denoted .
If is not in the column space of , then the closest you can come to solving is to solve .
Since , you know that is one solution to .
And since , all solutions are given by .
Use the graphic to verify that the vectors are solutions to for every value of .
Modified three steps to solving .
Step 1: From the plot you see .
Since , throw away the portion of to get .
Continue solving .
Step 2: Divide the by the singular value 2 to get .
Step 3: Form by using , i.e. .
Since , the general solution to is .
This general solution to is called the general least squares solution to .
(Q31) The plot shows that the singular values are and .
(Q32) The plot shows .
(Q33) The vector in the column space of closest to is .
(Q34) How many solutions does have? ___
How many solutions will have? ___
(Q35) The general solution to is .
(Q36) The general least squares solution to is .
Copyright © 2007 Todd Will
Created by Mathematica (April 14, 2007)