Geometry and algebra of linear maps, vectors, and matrices
A linear map x ↦ Wx always sends the zero vector to zero, since W·0 = 0. That is why a purely linear transformation keeps the origin fixed, no matter how much it rotates, stretches, or reflects everything else. Plenty of useful maps need one more move: shift the whole transformed space by a fixed vector, so the origin can land somewhere else entirely.
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▶ Affine Maps & the Bias Term