Science:Math Exam Resources/Courses/MATH307/April 2006/Question 05 (a)
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Question 05 (a) 

A matrix A and a vector b are given by
Find an orthonormal basis for the column space of A (i.e. for R(A)). Express the matrix A in the form A = QR, where Q is a matrix with orthonormal columns and R is upper triangular and with positive diagonal entries. 
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Hint 

Science:Math Exam Resources/Courses/MATH307/April 2006/Question 05 (a)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. If we find the row reduced echelon form we get that
Take the pivot columns to be our column space
We can then find the orthonormal basis let and let be the orthonormal vectors we wish to find
=
where Q is the matrix formed by the orthonormal basis and R is upper triangular with position diagonal entries
Solving yields
