Tag: projection

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P = P. It leaves its image unchanged. Though abstract, this definition of “projection” formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.
For example, the function which maps the point (x, y, z) in three-dimensional space R to the point (x, y, 0) is a projection onto the x-y plane. This function is represented by the matrix
The action of this matrix on an arbitrary vector is
To see that P is indeed a projection, i.e., P = P, we compute:
A simple example of a non-orthogonal (oblique) projection (for definition see below) is
Via matrix multiplication, one sees that
proving that P is indeed a projection.
The projection P is orthogonal if and only if = 0.
For simplicity, the underlying vector spaces are assumed to be…