Affine mapping/Basis/Image points/Mapping properties/Exercise

Let ${}E$ and ${}F$ be affine spaces over the field ${}K$, let ${}P_{1},\ldots ,P_{n}\in E$ be an affine basis of ${}E$, and let ${}Q_{1},\ldots ,Q_{n}\in F$ denote points. Let

$\psi \colon E\longrightarrow F$

be the corresponding affine-linear mapping with

${}\psi (P_{i})=Q_{i}\,.$

Show the following statements.

a) ${}\psi$ is bijective if and only if ${}Q_{1},\ldots ,Q_{n}$ is an affine basis of ${}F$.

b) ${}\psi$ is injective if and only if ${}Q_{1},\ldots ,Q_{n}$ are affinely independent.

c) ${}\psi$ is surjective if and only if ${}Q_{1},\ldots ,Q_{n}$ is an affine generating system of ${}F$.

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