Least Squares Solutions

Here's the picture of for the matrix .

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 .

Here's the picture of for the matrix .

(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 .

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Copyright © 2007 Todd Will

Created by
Mathematica (April 14, 2007)