Affine transformations & Affine spaces

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Thanks Benjamin:

I am aware how it could be used in vector algebra. The source I use is Geometric Tools for Computer Graphics by Schneider & Eberly 2003.

There is a distinction unlike vector algebra between a vector and a point. *Add two vectors to get a third. *Multiply a vector by a scalar to get a vector. *Add a vector to a point to get another point. *Subtract two points to get vector. (edited) *etc.

In affine algebra, a point can be translated in a matrix multiplication where in vector algebra it cannot. I have to admit that vector algebra is used quite a lot in CG, for example, the affine matrix has a vector matrix embedded.

The elements of a point in affine space for example is [0, 1, 0, 1] for the vector [0, 1, 0] in vector space. The elements for a vector in affine space is [0, 1, 0, 0] for the vector [0, 1, 0] in vector space.

Please don't ask me to prove this. I do believe that sage developers can implement this as a subset to the module linear algebra. CG is a large and fast growing business.

I suppose the answer from Benjamin is NO. There is no affine algebra unless we define the vectors as points and the matrices as affine matrices....

Many Thanks