Can't you simply turn affine functions into linear ones by shifting the origin of your coordinate system?
Max
Not really - affine functions aren't just operating on vectors in a 'shifted' coordinate frame, because 'shifting' itself is not a linear transformation (try representing it as a matrix!). If you consider g(x) = x-b/a (shifting the origin), f(g(x)) is still non-linear because g is not linear: f(g(x+y)) != f(g(x)+g(y)). The problem is really that affine functions add the 'shift' at every invocation, and it doesn't make sense for a 'linear' result to depend on how you decompose your vector.
However, what we can do to replace affine functions with linear ones is use homogeneous coordinates, which will be covered in a few lectures.
Can't you simply turn affine functions into linear ones by shifting the origin of your coordinate system?
Not really - affine functions aren't just operating on vectors in a 'shifted' coordinate frame, because 'shifting' itself is not a linear transformation (try representing it as a matrix!). If you consider g(x) = x-b/a (shifting the origin), f(g(x)) is still non-linear because g is not linear: f(g(x+y)) != f(g(x)+g(y)). The problem is really that affine functions add the 'shift' at every invocation, and it doesn't make sense for a 'linear' result to depend on how you decompose your vector.
However, what we can do to replace affine functions with linear ones is use homogeneous coordinates, which will be covered in a few lectures.