Why does affine transformation need an addition of vector in matrix representation, while linear does not?

The linear transformation preserves addition and multiplication by scalar, and can be performed by matrix multiplication

Ax=b,

where b is transformed vector, x - original - and A's columns are transformed basis vectores of the original vector space.

The affine transformation preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. As it maps from/to affine space, it also doesn't need to preserve the origin.

It's matrix representation is Ax+b=c, where c is the transformed vector and A - matrix of linear transformation.

Can somebody explain me the significance of the addition of b vector (a translation after linear transform)?

• Without the added vector, the origin always maps to the origin, because $A0 = 0$.
– user856
Jul 15, 2018 at 14:47