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Unformatted text preview: Lecture 26 Andrei Antonenko April 11, 2003 1 Orthogonality Let V be a Euclidean space, and let v and u be 2 vectors in this space. Then we can define the angle between these 2 vectors. Definition 1.1. The angle between two vectors v and u from the vector space V can be defined by the following formula: cos = h u,v i k u kk v k Actually, we should check that this definition is correct we have an expression for cosine, and it should belong to the interval [ 1 , 1]! It is easy to check. From CauchyBunyakovsky Schwartz inequality we have h u,v i k u kk v k , and so we will have 1 h u,v i k u kk v k 1 Lets note that this is a general definition which works in many different spaces, and not only in R 2 and R 3 . For example, by this formula we can define the angle between two functions from C [ a,b ] or between two polynomials. Example 1.2. Let u = (1 , 2 , 3) and let v = ( 1 , 2 , 2) . Then h u,v i = 1 + 4 6 = 4 , k u k = 1 + 4 + 9 = 14 , and k v k = 1 + 4 + 4 = 3 . So, the angle between these two vectors can be defined by the following formula: cos = h u,v i k u kk v k = 4 3 14 . Another important concept is a normalization of the vector. Definition 1.3. If v is a vector from the vector space V then the vector v k v k is called a normalization of v . 1 The main property of normalization is that its norm is equal to 1. So, we take a vector which is proportional to v with the length 1. Now well give the very important definition main definition of this lecture. Definition 1.4. Two vectors u and v are called orthogonal if h u,v i = 0 . Let we have a vector v from the Euclidean space V . Lets consider all vectors u which are orthogonal to v , i.e. the set of vectors u such that h v,u i = 0: v = { u V h u,v...
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 Spring '03
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