Solution to Exercise 3.20.
I cannot teach you. I can help you to explore and nothing more.
—Bruce Lee (1940–1973)

Exercise 3.20. Prove Proposition 3.3. An affine transformation of the Euclidean plane is a transformation. 

The proposition is showing that the term affine transformation is well-defined.

Proof. Let A be the matrix for an affine transformation of the Euclidean plane T. Since A is the matrix of an affine transformation, the determinant of A is nonzero. Hence, the inverse of A exists.
        We first show T is a one-to-one function. Let X and Y be points in the Euclidean plane. Assume T(X) = T(Y). Thus, AX = AY. Hence, X = A^–1(AX) = A^–1(AY) = Y. Therefore, T is a one-to-one function.
        Next, we show T maps the Euclidean plane onto the Euclidean plane. Let Y be an arbitrary point in the Euclidean plane. Set X = A^–1Y. Then T(X) = AX = A(A^–1Y) = Y. Hence, T maps the Euclidean plane onto the Euclidean plane.
        Therefore, since T is a one-to-one function that maps the Euclidean plane onto the Euclidean plane, T is a transformation.//