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In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspace of a projection is also called a projection, even if the idempotence property is lost. An everyday ...
A square matrix is called a projection matrix if it is equal to its square, i.e. if =. [2]: p. 38 A square matrix is called an orthogonal projection matrix if = = for a real matrix, and respectively = = for a complex matrix, where denotes the transpose of and denotes the adjoint or Hermitian transpose of .
Projection (mathematics) – Mapping equal to its square under mapping composition; Projection (measure theory) Projection (linear algebra) – Idempotent linear transformation from a vector space to itself; Projection (relational algebra) – Operation that restricts a relation to a specified set of attributes
The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b.
Projection (mathematics), any of several different types of geometrical mappings Projection (linear algebra), a linear transformation P from a vector space to itself such that P 2 = P; Projection (set theory), one of two closely related types of functions or operations in set theory; Projection (measure theory), use of a projection map in ...
If 0° ≤ θ ≤ 90°, as in this case, the scalar projection of a on b coincides with the length of the vector projection. Vector projection of a on b (a 1), and vector rejection of a from b (a 2).
A matrix, has its column space depicted as the green line. The projection of some vector onto the column space of is the vector . From the figure, it is clear that the closest point from the vector onto the column space of , is , and is one where we can draw a line orthogonal to the column space of .
A parallel projection is a particular case of projection in mathematics and graphical projection in technical drawing. Parallel projections can be seen as the limit of a central or perspective projection, in which the rays pass through a fixed point called the center or viewpoint, as this point is moved towards infinity.