 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2021/digital-systems-linear-algebra-introduction-to-formal-proof/6/b/solution

A unit vector along the line is

d = (3, 4)^{⊤} /5

. Reflection preserves the orthogonal projection

d d^{⊤} u

onto that line and negates its perpendicular component. Therefore

A u = d d^{⊤} u - (u - d d^{⊤} u)

and

A = 2 d d^{⊤} - I = \frac{1}{25} (- 7 24 247) .

As checks,

A (3, 4)^{⊤} = (3, 4)^{⊤}

A (- 4, 3)^{⊤} = - (- 4, 3)^{⊤}

, and

A^{2} = I

. Its determinant is minus one, as expected for a planar reflection matrix.

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