Intuition: Affine map vs Linear map In my Geometry course, we defined "affine map" by saying it's a map $T$ of the form $T(\mathbf{x}) =  L(\mathbf{x})+\mathbf{b}$, where $L$ is a linear map. So an affine map is just a linear map with a translation by $T(\mathbf{0})=\mathbf{b}$.
Questions:

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*Is any map that preserves collinearity (ie. it maps a line to a line) affine?


*If it's just a linear map with a translation, why bother studying it as a separate object? Instead of focusing all our attention on general linear maps.
 A: 
Is any map that preserves collinearity (ie. it maps a line to a line) affine?

No, not every collineation is an affine transformation.
According to the fundamental theorem of projective geometry, every collineation is a combination of a projective transformation and an automorphism of the underlying field. If the underlying field is the real numbers, then there is no non-trivial automorphism so every collineation over the reals is a projective transformation.
Projective transformations in general don't have to preserve parallelism. They form a larger class of transformations than the affine transformations. Every projective transformation that preserves parallel lines is an affine transformation.
Also, projective transformations are typically expressed as linear transformations with one more dimension, acting on homogeneous coordinate vectors.
One potential caveat is that the concept of a collineation is typically expressed for a projective plane, which in addition to the usual lines has one extra line, the line at infinity. Affine transformations leave the line at infinity invariant. Other projective transformations map a finite line to the line at infinity. If you restrict your view to the affine plane, then you could say that such a transformation does not really map lines to lines, because there is one line that doesn't get mapped to a line of the affine plane. If you take this stance and only consider transformations which map all lines of the affine plane to lines of the affine plane, without even a single exception and without allowing for the existence of a line at infinity, and if your affine plane is over the real numbers, then yes, those transformations would all be affine transformations.

If it's just a linear map with a translation, why bother studying it as a separate object? Instead of focusing all our attention on general linear maps.

If you have a wider class of transformations, you can use them to describe a wider range of operations that might have practical relevance for the field you study. If for example you are doing some animation, then being able to move things around would be very important. I don't see how a study of linear transformations could be a substitute for that.
A: Regarding question 2, here is an example which I hope will be sufficiently telling.
In vector space $E=\mathbb{R}^2$ :
Let $O=(0,0), {\color{green}{O_1=(6,3)}}, s_O:E\to E, (x,y)\mapsto (-x,-y)$ and $s_{{\color{green}{O_1}}}:(x,y)\mapsto (12-x,6-y)$. Then, $s_{{\color{green}{O_1}}}s_O=t_{2{\color{red}b}}$, with ${\color{red}b}=O_1-O=(6,3)$. $s_O$ is is a well-known linear map; $s_{O_1}$ is an affine map ($s_{O_1}=t_{2b}s_O$).

