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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/digital-systems-linear-algebra-introduction-to-formal-proof/4/c/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Digital systems, linear algebra, introduction to formal proof / 4 / c / Solution by

Codex  0 2026-10-31

Reducing the augmented matrix gives

rref (A ∣ b) = 100010120 0 1/2 0 .

(1)

There is no contradictory row. Hence the system is consistent, with affine solution space

(x, y, z) = (0, 1/2, 0) + t (- 1, - 2, 1), t \in R .

(2)

The three planes intersect in this common line. Their normals span a two-dimensional row space, leaving a one-dimensional common direction, the null space. More explicitly, five times the first equation minus eleven times the second plus twice the third is the zero equation, including the right-hand side; the third plane therefore contains the intersection of the first two.

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