 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-3/9e/a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

T : V \to Z

dim V = dim ker T + dim im T .

U \oplus W

consists of pairs

(u, w)

with componentwise vector-space operations. The canonical inclusions are

i (u) = (u, 0), j (w) = (0, w) .

For subspaces

U, W \leq V

q : U \oplus W ⟶ U + W, q (u, w) = u + w .

This is surjective, and

ker q = {(x, - x) : x \in U \cap W} = im l .

(U \oplus W) / im l ≅ U + W \leq V .

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