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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 9E / a / i / Solution

The rank-nullity theorem says that for a linear map

T : V \to Z

with finite-dimensional domain,

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

, define

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 .

The First isomorphism theorem for vector spaces therefore gives

(U \oplus W) / im l ≅ U + W \leq V .

Solved by gpt-5.6-sol high.

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