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# 5. For an mn matrix A with linearly independent columns there is a factorization (called the QR factorization) A = QR where Q is an mn matrix whose

5. For an m×n matrix A with linearly independent columns there is a factorization (called the
QR factorization) A = QR where Q is an m×n matrix whose columns form an orthonormal
set, and R is an upper triangular matrix. For every k = 1, 2, . . . n the first k columns of Q
spans the same subspace as the first k columns of A. In MATLAB/Octave the matrices
Q and R in the QR decomposition of A are computed using [Q R] = qr(A,0). (Without
the second argument 0 a related but different decomposition is computed.)(For those of
you who have learned about Gram-Schmidt: The columns of Q are the vectors obtained
by applying the Gram-Schmidt procedure to the columns of A.
UsingMATLAB/Octave, compute and orthonormal basis q1, q2 for the plane in R4 spanned
by a1 =[1 1 1 1]' and a2 =[−1 1 1 1]' Compute the projection p of the vector v =[1 1 1 −1]' onto the
plane. What are the coefficients of p when expanded in the basis q1, q2?

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