projectionsthat

Projectionsthat is not a recognized term in standard mathematical or computational literature. As written, it appears to be a concatenation of “projections” and “that” and does not denote a specific concept.

In general usage, the phrase “projections that” would introduce a clause describing which projections have certain properties. In mathematics, a projection operator P on a vector space V is a linear transformation satisfying P^2 = P. Its image is a subspace Im(P) and its kernel is Ker(P). If an inner product is given, a projection is often orthogonal, meaning P is self-adjoint with respect to that inner product and maps vectors to their closest element in a subspace. Projection matrices represent such operators with respect to a basis, and V decomposes as the direct sum of Im(P) and Ker(P).

In statistics, projection concepts appear in linear models, where fitted values are obtained by projecting the observed data onto the column space of the design matrix. The hat matrix H = X(X^T X)^{-1} X^T projects vectors of responses onto the space of fitted values.

In computer graphics, projection refers to transforming 3D coordinates into 2D screen coordinates using perspective or orthographic projection matrices.

Usage note: “projectionsthat” as a single word is unusual and could be a typographical error; the standard form is “projections that.”

See also: projection (mathematics), projection matrix, orthogonal projection, perspective projection.