Transformation vs projection matrix, what are the differences by definition

Question

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?

Answer 1

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.

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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).

Answer 2

"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 $P^2=P$.