Let \(\displaystyle \mathbb{A}_1\) and \(\displaystyle \mathbb{A}_2\) be affine lines. Let \(\displaystyle x\) be an affine coordinate function on \(\displaystyle \mathbb{A}_1;\) let \(\displaystyle y\) be an affine coordinate function on \(\displaystyle \mathbb{A}_2.\) Let \(\displaystyle f:\mathbb{A}_1\, \rightarrow\, \mathbb{A}_2\) be an affine mapping. Associated with \(\displaystyle f\) is a function \(\displaystyle F:\mathbb{R}\, \rightarrow\, \mathbb{R}\) such that if \(\displaystyle Q\, =\, f(P),\) then \(\displaystyle y(Q)\, =\, F\, \circ\, x(P).\) Show that the most general formula for \(\displaystyle F\) is \(\displaystyle F(\alpha)\, =\, r\alpha\, +\, s.\)
As the book says , an affine function of a line is
and represent the real number that, multiplied for a basis and starting from an origin of the line gives a certain point of the line, so a origin of the line and a basis is implicitly taken when defining the affine coordinate function.
Suppose we have 2 affine lines (NOT paralells), for each line we set an arbitrary origin and an arbitrary basis. I choose a point P in the first line and i get x(P) = 3. Then, I choose an arbitrary point P' in the second line , and i get y(P') = 1 . So, the mapping beetween the first line and the second line is F(α)=α/3.
Why it's wrong and F(α) should be F(α)=rα+s, and not just F(α)=rα as my result ?
As the book says , an affine function of a line is
A→R
and represent the real number that, multiplied for a basis and starting from an origin of the line gives a certain point of the line, so a origin of the line and a basis is implicitly taken when defining the affine coordinate function.
Suppose we have 2 affine lines (NOT paralells), for each line we set an arbitrary origin and an arbitrary basis. I choose a point P in the first line and i get x(P) = 3. Then, I choose an arbitrary point P' in the second line , and i get y(P') = 1 . So, the mapping beetween the first line and the second line is F(α)=α/3.
Why it's wrong and F(α) should be F(α)=rα+s, and not just F(α)=rα as my result ?
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