5.2 gram-schmitt process the first 2 steps of the gram schmitt process for more information...
TRANSCRIPT
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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
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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.
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A Geometric view
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Example 1a,b
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1a
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1b Solution
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The Gram-Schmidt Process
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Problem 6
Perform the Gram-Schmitt process
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Problem 6 Solution
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Problem 4
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Solution to problem 4
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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
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Problem 16
• Find the QR factorization of the following matrix
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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=
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Problem 20
Find the QR factorization of
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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
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Problem 21Find the QR factorization
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Problem 21 solutionM=
M = QR therefore Q-1M = R
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Homework p.209 1- 31 odd
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