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The transformation P is the orthogonal projection onto the line m. In linear algebra and functional analysis , a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism ) such that P ∘ P = P {\displaystyle P\circ P=P} .
The projection of a onto b is often written as or a ∥b. The vector component or vector resolute of a perpendicular to b, sometimes also called the vector rejection of a from b (denoted or a ⊥b), [1] is the orthogonal projection of a onto the plane (or, in general, hyperplane) that is orthogonal to b.
A general approach to the least squares problem ‖ ‖ can be described as follows. Suppose that we can find an n by m matrix S such that XS is an orthogonal projection onto the image of X. Then a solution to our minimization problem is given by
This point y is the orthogonal projection of x onto F, and the mapping P F : x → y is linear (see § Orthogonal complements and projections). This result is especially significant in applied mathematics, especially numerical analysis, where it forms the basis of least squares methods. [74]
For example, the mapping that takes a point (x, y, z) in three dimensions to the point (x, y, 0) is a projection. This type of projection naturally generalizes to any number of dimensions n for the domain and k ≤ n for the codomain of the mapping. See Orthogonal projection, Projection (linear algebra). In the case of orthogonal projections ...
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 .
Orthographic projection (also orthogonal projection and analemma) [a] is a means of representing three-dimensional objects in two dimensions.Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection plane, [2] resulting in every plane of the scene appearing in affine transformation on the viewing surface.
Orthogonal projection onto a line, m, is a linear operator on the plane. This is an example of an endomorphism that is not an automorphism. In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism.