And what this means geometrically is that the translation becomes a shear in the 4d space, which is a linear transform. It’s easier to visualize if you think in 2D; the matrix

  (1 b) 
  (0 1)

maps a point (x, 1) to (x+b, 1) which is exactly what we want, while vectors (which we model as points (x, 0)) remain at (x, 0) as they should - translation shouldn’t act on vectors!