5.2 Gram-Schmitt Process

The first 2 steps of the Gram Schmitt ProcessFor more information visit:http://en.wikipedia.org/wiki/Gram

%E2%80%93Schmidt_process

It is often helpful to take a basis of a subspace and write it in a form such

that the vectors are orthonormalTo do this just subtract off the component of

the vector that is parallel to the previous vectors in the basis. Repeat this process for each vector in the basis.

A Geometric view

Example 1a,b

1a

1b Solution

The Gram-Schmidt Process

Problem 6

Perform the Gram-Schmitt process

Problem 6 Solution

Problem 4

Solution to problem 4

QR factorizationList the columns of the orthornomal matrix found in Gram-Schmidt Then use the formula below to find a matrix R that when multiplied by Q generates M (the original matrix)This factoring has some applications in higher math. Note: This formula works

in 2 D, R can be seen as a series of coordinate Vectors we will use this for higher dimensions

Problem 16

• Find the QR factorization of the following matrix

Solution to 16

Note: the vectors are already orthogonal so only divide by the length.Create R either by formula or by coordinate vectors.

M=

Problem 20

Find the QR factorization of

Problem 20 solutionM=

One shortcut. If rref of M is I then I is an orthonormal basis of M and Gram-Schmitt is not required

Problem 21Find the QR factorization

Problem 21 solutionM=

M = QR therefore Q-1M = R

Homework p.209 1- 31 odd