Least Squares Solutions by the numbers

Here's the picture of for the matrix .

You see that and .

Since you know that every vector gets sent to a multiple of .

Your job is to solve .

Step 1: Find the coordinates for .

From the plot it appears that .

Compute the exact coordinates:

.

Since , throw away the portion of to get .

Continue solving .

Step 2: Divide by the singular value 2 to get .

Step 3: Use this number as the coordinate for to get .

To get a numerical solution multiply out: .

So one solution to is .

Since , all solution to are given by

.

Final answer: There are NO solutions to . But the least squares solutions are .

Here's the picture of for the matrix .

Your job is to solve .

Step 1: Find the coordinates for .

From the plot it appears that .

Since is an orthonormal basis, you can compute the exact coordinates by using dot products.

(Q37)

Since , the projection of on the column space of is

(Q38)

(Q39) How many solutions will have? ___

How many solutions will have? ___

Step 2: Divide by the singular value.

Step 3: All least squares solution to are given by

(Q40)

(Q41) Multiply out to get a numerical answer. There are NO solutions to . But the least squares solutions are

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Copyright © 2007 Todd Will

Created by
Mathematica (April 15, 2007)