In this exercise, we establish a useful matrix representation for affine functions. We identify Rn with the n-dimensional affine subspace (as in Exercise 2.2.30)

consisting of vectors whose last coordinate is fixed at xn+1 = 1.
(a) Show that multiplication of vectors

by the (n + 1) × (n + 1) affine matrix

coincides with the action (7.32) of an affine function on x ˆˆ Rn.
(b) Prove that the composition law (7.34) for affine functions corresponds to multiplication of their affine matrices.
(c) Define the inverse of an affine function in the evident manner, and show that it corresponds to the inverse affine matrix.

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