5.2 Gram-Schmitt Process The first 2 steps of the Gram Schmitt Process For more information...
-
Upload
shana-osborne -
Category
Documents
-
view
223 -
download
0
Transcript of 5.2 Gram-Schmitt Process The first 2 steps of the Gram Schmitt Process For more information...
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