# Transformation vs projection matrix, what are the differences by definition

Is there a defined difference between the terms "projection" and "transformation" matrix? Is it that e.g., a transformation preserves the vector space of the object being transformed whereas projection can also imply projecting something into a new vector space with different basis vectors?

A projection is a special kind of transformation that

• goes from one space to itself
• satisfies $$P^2=P,$$ in the sense that applying it twice makes no difference.

Note that if the domain and codomain were unrelated, it would make no sense to apply it twice.

A second difference given the wording used in the question, is that a transformation is not intrinsically a matrix. A transformation can be represented by a matrix, if you have specified a basis for both the domain and codomain (which could be a different basis even if the two spaces are the same).

• Ah thanks, that makes sense. So, regarding the second part of your answer, an example of a transformation would be simply a scaling by a factor (scalar). Say factor a, then a*v, then v would still be in the same vector space with the same basis vectors but just scaled by a factor or a. – user83266 Jul 27 '18 at 18:29
• @Sebastian Yes. Regarding the comment you made on the other answer, you're wrong. A linear map is called a transformation even with different dimensions. Whereas for a projection matrix, you need a square matrix if you want to multiply it by itself (to check that $P\times P=P$). – Arnaud Mortier Jul 27 '18 at 19:42
• Awesome, thanks for clearing that up! – user83266 Jul 28 '18 at 0:58

"Transformation" can be almost anything. "Projections" map everything into a (usually) smaller subspace in which nothing changes. So a projection acts like the identity on its image. Usually this is written as $P\circ P=P$ or $P^2=P$.

• Thanks, so basically a projection matrix would be a special case of a transformation matrix. If I have a mxn dimensional matrix A and an nx1 dimensional vector v, then A is a linear transformation matrix if m=n, and probably some sort of projection otherwise – user83266 Jul 27 '18 at 18:26