6.2 - MATH1850U/2050U: Chapter 6 cont INNER PRODUCT SPACES...

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MATH1850U/2050U: Chapter 6 cont… 1 INNER PRODUCT SPACES cont… Angle and Orthogonality in Inner Product Spaces (6.2; pg. 345) Recall: In sections 3.2-3.3, you already learned about the Cauchy-Schwartz inequality, properties of lengths and distances, cosine of an angle between 2 vectors, and the Pythagorean Theorem for R n . Definition: Two vectors u and v in an inner product space are called orthogonal if 0 , v u . Example: Suppose p , q , and r are defined as given below. i) Are p and q orthogonal? ii) Are p and r orthogonal? Definition: Let W be a subspace of an inner product space V . A vector u in V is said to be orthogonal to W if it is orthogonal to every vector in W . The set of all vectors in V that are orthogonal to W is called the orthogonal complement of W . This set is denoted W .
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MATH1850U/2050U: Chapter 6 cont… 2 Theorem (Properties of Orthogonal Complements): If W is a subspace of an inner product space V , then: a) W is a subspace of V . b)
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6.2 - MATH1850U/2050U: Chapter 6 cont INNER PRODUCT SPACES...

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