Orthogonal Projection

For a given vector subspace M spanned by an orthonormal basis, we can define an orthogonal projection that maps any vector to a vector in the subspace M. Using an orthogonal projection, we can decompose any vector into two parts: one that belongs to M, the other that belongs to the orthogonal complement of M. Orthogonal projections have various interesting (and useful) properties including a generalized version of the Pythagorean theorem.
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