Orthogonality in Euclidean Space: The Pythagorean Theorem and the Orthogonal Decomposition

We prove the Pythagorean Theorem and derive the Orthogonal Decomposition in high dimensional Euclidean Space. The bilinear structure of the inner product together with the definition of orthogonality will help us understand the geometric importance of an inner product on a more general space. The orthogonal decomposition is equally important, as it will allow us to define projection onto a subspace. These projection operators will be essential to our study of Least Squares in Statistics and the Spectral Theorem.
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