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An orthogonal projection is a projection for which the range and the kernel are ... This expression generalizes the formula for orthogonal projections given above.
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. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
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.
The projection matrix has a number of useful algebraic properties. [5] [6] In the language of linear algebra, the projection matrix is the orthogonal projection onto the column space of the design matrix . [4]
The scalar projection is a scalar, equal to the length of the orthogonal projection of ... The formula above can be inverted to obtain the angle, ...
The vector is then defined to be the difference between and this projection, guaranteed to be orthogonal to all of the vectors in the subspace . The Gram–Schmidt process also applies to a linearly independent countably infinite sequence { v i } i .
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, the space admits a decomposition as a product, and the projection operator is a projection in that sense as well.
Hilbert projection theorem — For every vector in a Hilbert space and every nonempty closed convex , there exists a unique vector for which ‖ ‖ is equal to := ‖ ‖. If the closed subset C {\displaystyle C} is also a vector subspace of H {\displaystyle H} then this minimizer m {\displaystyle m} is the unique element in C {\displaystyle C ...