Orthogonal projections - Orthogonality and Least Squares
Orthogonal projections
Lessons
Notes:
The Orthogonal Decomposition Theorem
Let be a subspace in . Then each vector in can be written as:
where is in and is in . Note that is the orthogonal projection of onto
If {} is an orthogonal basis of , then
proj
However if {} is an orthonormal basis of , then
proj
Property of Orthogonal Projection
If {} is an orthogonal basis for and if happens to be in , then
proj
In other words, if y is in Span{}, then proj.
The Best Approximation Theorem
Let be a subspace of . Also, let be a vector in , and be the orthogonal projection of onto . Then is the closest point in , because
<
where are all vectors in that are distinct from .
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Intro Lesson
Orthogonal Projections Overview:
