6.3 - MATH1850U/2050U Chapter 6 cont 1 INNER PRODUCT SPACES...

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MATH1850U/2050U: Chapter 6 cont. .. 1 INNER PRODUCT SPACES cont. .. Gram-Schmidt Process; QR-Decomposition (6.3; pg. 352) Definition: A set of vectors in an inner product space is called an orthogonal set if all pairs of distinct vectors in the set are orthogonal. An orthogonal set in which each vector has norm 1 is called orthonormal . Recall: Previously, we saw that given any vector u , then v defined by || || u u v is a unit vector (that is, it has norm 1) and is a scalar multiple of u (parallel to u ). Example: Consider the set of vectors shown below. i) Show that the vectors form an orthogonal set in R 3 with the Euclidean inner product. ii) Construct an orthonormal set from these vectors. Definition: A basis of an inner product space that consists of orthonormal vectors is called an orthonormal basis . A basis of an inner product space consisting of orthogonal vectors is called an orthogonal basis .
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MATH1850U/2050U: Chapter 6 cont. .. 2 Examples: Theorem: If  n S v v v , , , 2 1 is an orthonormal basis for an inner product space V , and u is any vector in V , then n n v v u v v u v v u u  , , , 2 2 1 1 Explanation/Example: Note: Because the components of the coordinate vector of
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This note was uploaded on 10/12/2011 for the course MATH 1020 taught by Professor Paulatu during the Spring '11 term at UOIT.

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6.3 - MATH1850U/2050U Chapter 6 cont 1 INNER PRODUCT SPACES...

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