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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