 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2017/digital-systems-linear-algebra-introduction-to-formal-proof/4/f/solution

Collect the orthonormal vectors as columns and compute

R = Q^{T} A

. This gives the thin QR decomposition

Q = \frac{1}{2} 1111 1 - 1 - 1 1 1 - 1 1 - 1, R = 200420222 .

The inner products in part (e) prove

Q^{T} Q = I_{3}

; the column identities

a_{1} = 2 q_{1}

a_{2} = 4 q_{1} + 2 q_{2}

a_{3} = 2 q_{1} + 2 q_{2} + 2 q_{3}

prove

A = QR

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