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If is a unit vector on the line, then the projection is given by the outer product =. (If is complex-valued, the transpose in the above equation is replaced by a Hermitian transpose). This operator leaves u invariant, and it annihilates all vectors orthogonal to u {\displaystyle \mathbf {u} } , proving that it is indeed the orthogonal ...
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 .
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 .
Since (+) = + (the property of pseudoinverse), the matrix is an orthogonal projection onto the image (column-space) of X. In accordance with a general approach described in the introduction above (find XS which is an orthogonal projection),
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]
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.
As with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation. Parallel projections are also linear transformations and can be represented simply by a matrix. However, perspective projections are not, and to represent these with a matrix, homogeneous coordinates can be ...
In either the coordinate or vector formulations, one may verify that the given point lies on the given plane by plugging the point into the equation of the plane. To see that it is the closest point to the origin on the plane, observe that p {\displaystyle \mathbf {p} } is a scalar multiple of the vector v {\displaystyle \mathbf {v} } defining ...