Affine mapping/Graph/Factorization/Exercise

Let

$\varphi \colon E\longrightarrow F$

be an affine-linear mapping between the affine spaces ${}E$ and ${}F$ over ${}K$.

a) Show that the graph ${}G$ of ${}\varphi$ is an affine subspace of the product space ${}E\times F$.

b) Show that the mapping

$\psi \colon E\longrightarrow G,P\longmapsto (P,\varphi (P)),$

is an isomorphism of affine spaces.

c) Show that

${}\varphi =p_{2}\circ \psi \,$

holds, where ${}p_{2}$ is the projection onto ${}F$.

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