Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 1 / 9F / ii

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- Solution ii

Solution

True. Choose a basis

v_{1}, \dots, v_{r}

of

V

. By surjective function, choose

u_{i} \in U

with

α u_{i} = v_{i}

. Define the linear map

β

by

β v_{i} = u_{i}

and extend by linearity. Then

α β v_{i} = v_{i}

on every basis vector, and therefore

α β = id_{V} .

This constructs a right inverse and does not require

α

to be injective. If

V

is the zero vector space, the unique zero map is the required right inverse.

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