Abstract
A proof of Cochran's theorem that emphasizes the geometry involved more than the usual proof is given below. The prerequisite background needed in algebra and geometry (inner product space theory) restrict its usefulness to graduate or advanced upper division courses. All vectors will be in n dimensional Euclidean space, Rn, and I v will denote the usual Euclidean length or norm. Theorem: Let Si, i = 1, ..., r be subspaces of Rn with dimensions ni, respectively, and ;ri., ni < n. Let Piv denote the orthogonal projection of v into Si. If there exist r real-valued functions fi with domain Rn such thatfi(0) = 0 for all i and 11 v 112 = i=1rf(Piv) for all v in Rn, then:
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