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Week_10 - Week 10 Section 5.3 Orthogonality We looked at...

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Week 10 Section 5.3: Orthogonality We looked at dot product and length of vectors in R 2 and R 3 in Chapter 4. We can extend these definitions to R n . 1. If u = [ u 1 u 2 . . . u n ] T and v = [ v 1 v 2 . . . v n ] T are vectors in R n , then u · v = u 1 v 2 + u 2 v 2 + . . . + u n v n . 2. Two vectors u and v R n are orthogonal if u · v = 0. 3. The length (or norm) of a vector u = [ u 1 u 2 · · · u n ] T is u = u 2 1 + u 2 2 + . . . + u 2 n and is always positive. 4. u 2 = u · u Cauchy Inequality If u, v R n , then | u · v | ≤ u v . Proof: The general case is proven in the textbook, we will consider the case where n = 2. In R 2 , we can use the definition of dot product involving cos θ to prove this. | u · v | = | u v cos θ | = u v | cos θ | u v (1), since | cos θ | ≤ 1 Triangle Inequality u + v u + v Proof: We will prove that u + v 2 ( u + v ) 2 , and since u + v 0 and u + v 0 we can take square roots of both sides and the desired inequality will hold. u + v 2 = ( u + v ) · ( u + v ) = u · u + u · v + v · u + v · v = u 2 + 2 u · v + v 2 u 2 + 2 | u · v | + v 2 since any real number is less than equal to its absolute value u 2 + 2 u v + v 2 by the Triangle Inequality = ( u + v ) 2 as desired. 1
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Definition: A set { u 1 , u 2 , . . . , u n } of non-zero vectors in R n is called an orthogonal set if u i · u j = 0 for all i = j . This set { u 1 , u 2 , . . . , u n } is called orthonormal if it is orthogonal and each u i is a unit vector. (ie. if u i = 1 for all i .) Examples 1. The standard basis { e 1 , e 2 , . . . , e n } is an orthonormal set in R n . 2. If { u 1 , u 2 , . . . , u n } is orthogonal, then so is { a 1 u 1 , a 2 u 2 , . . . , a n u n } . We can create an orthonormal set from any orthogonal set simply by dividing each vector by its length making it a unit vector. That is, if { u 1 , u 2 , . . . , u k } is an orthogonal set, then 1 u 1 u 1 , 1 u 2 u 2 , . . . , 1 u k u k is an orthonormal set. Example: If f 1 = 1 1 1 - 1 , f 2 = 1 0 1 2 , v 3 = - 1 0 1 0 , and f 4 = - 1 3 - 1 1 , then show that { f 1 , f 2 , f 3 , f 4 } is orthogonal then normalize this set. Pythagorean Theorem in R n If { u 1 , u 2 , . . . , u n } is orthogonal, then u 1 + u 2 + . . . + u k 2 = u 1 2 + u 2 2 + . . . + u k 2 2
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Proof: In R 2 or R 3 , we can see the connection between orthogonal sets and linear dependence. Two orthogo- nal vectors in R 2 are linearly independent and three orthogonal vectors in R 3 are linearly independent.
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