04-11-03

# 04-11-03 - Lecture 26 Andrei Antonenko 1 Orthogonality Let...

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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 θ between two vectors v and u from the vector space V can be defined by the following formula: cos θ = 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 Cauchy-Bunyakovsky- 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 Let’s 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 θ between these two vectors can be defined by the following formula: cos θ = 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 it’s norm is equal to 1. So, we take a vector which is proportional to v with the “length” 1. Now we’ll 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 . Let’s 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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04-11-03 - Lecture 26 Andrei Antonenko 1 Orthogonality Let...

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