Which is the correct answer? Let A be an m x n matrix with linearly independent

Question

Which is the correct answer?

Let A be an m x n matrix with linearly independent columns and let A = QR be a QR factorization of A. Show that A and Q have the same column space. Since Q has orthonormal columns, it follows that col(A) = col(R), which implies col(A)-Col(Q). Since Q has orthonormal columns, it follows that row(A) row(R), which implies col(A)-col(Q) since A = QR, AT-RTQT, QT has orthonormal rows since Q has orthonormal columns, which implies col(A) = col(Q) O Since A = QR, A. RTQT, RT is invertible since R is invertible, so col(AT)-col(QT), which implies col(A) = col(Q) Since A = OR, AT RTQT. RT is invertible since R is invertible, so row(AT)-row(QT), which implies col(A)-Col(Q)

Explanation / Answer

Option (e) is correct.

Note that Q need not have orthonormal columns, so the first three options are wrong.

Also R has n linealy independent columns, hence R is invertible.

Also the implication in option 4 is not always true as col (AT) need not equal col (A).

But the implication in option 5 is lways true as row (AT) equals col (A).

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