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,

The direct sum $U\oplus W$ consists of pairs $(u,w)$ with componentwise vector-space operations. The canonical inclusions areFor subspaces $U,W\le V$, defineThis is surjective, andThe First isomorphism theorem for vector spaces therefore gives