 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-4/19j/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Write

u (x) = (10 x 1) .

Since

u (x) u (y) = u (x + y)

χ (v \cdot) (u (x + y)) = χ (u (vx + v y)) = χ (u (vx)) χ (u (v y)) = χ (v \cdot) (u (x)) χ (v \cdot) (u (y)) .

Thus $χ (v \cdot)$ is a one-dimensional representation of $U$ ; for $v = 0$ it is the trivial representation.

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