The underlying linear map of an affine map is unique

up:: Affine Map

The underlying linear map $L:V_M\to V_N$ of an affine map $f:A_M\to A_N$ is unique. Suppose otherwise: let $L,\tilde{L}:V_M\to V_N$ underlying linear maps for the same $f$. Then

$$f(A+v)=f(A)+L(v)=f(A)+\tilde{L}(v)$$

However, since an affine space is defined by a Regular Group Action, any pair of points in it are connected by a unique vector. Thus

$$\forall v\in V_M,L(v)=\tilde{L}(v)⟹L=\tilde{L}$$

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References

• Notes on Mathematical Physics for Mathematicians - Daniel Tausk