Generalize to matrices

You've seen that each matrix has an orthonormal basis which it stretches and sends to an orthonormal basis .

Analogous facts hold in higher dimensions.

A matrix has an orthonormal basis for which it stretches and sends to an orthonormal basis for .

A matrix has an orthonormal basis for which it stretches and sends to an orthonormal basis of . (The vector has singular value and is sent to the zero vector .)

A matrix has an orthonormal basis for which it stretches and sends to the first two vectors of an orthonormal basis of .

Column space and nullspace.

For a matrix you saw that the column space of was all multiples of and that the nullspace of was all multiples of .

In general for an matrix , the vectors with non-zero singular values are a basis for the column space of and the vectors with zero singular values are a basis for the nullspace of .

Suppose a matrix has singular values , , and .

The singular value of means that sends multiples of to the zero vector and that sends every to the plane spanned by and .

The set is a basis for the column space and the set is a basis for the nullspace.

Modified three steps to solving .

Step 1: Find so that .

Since , the portion of is inaccessible.

Throw away the portion and keep just .

Continue solving .

Step 2: Divide by the singular values: .

Step 3: Use these numbers to form .

The vector is a solution to and a least squares solution to .

Since , the general least squares solution to is .

(Q42) Suppose . In terms of the and or :

a. Give a basis for the nullspace of .

b. Give a basis for the column space of .

c. Give the general least squares solution to .

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Copyright © 2007 Todd Will

Created by
Mathematica (April 15, 2007)