MATH307 April 2006
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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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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 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