 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof-resit/6/b/iii/solution

The supplied factors give

Q^{T} b = (0, 2)^{T}

R x = Q^{T} b

yields

2 y = 2

and

10 x + 4 y = 0

, so

x = - 2/5, y = 1.

Multiplying by the original matrix gives

(1, 1, 1, 1)^{T}

, so this least-squares minimizer has residual zero: this particular overdetermined system is exactly consistent. Full column rank still guarantees uniqueness of the minimizer.

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