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Unformatted text preview: ENEE 241 02* READING ASSIGNMENT 9 Thu 03/05 Lecture Topics: inner products, norms and angles; projection Textbook References: sections 2.10.1–2.10.4 Key Points: • The norm, or length, of a realvalued vector a is given by k a k = h a , a i 1 / 2 , where h· , ·i denotes inner (dot) product. • The angle θ between two vectors a and b satisfies cos θ = h a , b i k a k · k b k • The (orthogonal) projection of b on a is the vector λ a such that b λ a is orthogonal to a , i.e., h b λ a , a i = 0. Therefore λ = h a , b i k a k 2 • The range (or column space) R ( V ) of the m × n matrix V = £ v (1) ... v ( n ) / consists of all linear combinations of columns of V , i.e., vectors of the form Vc . • The projection of a mdimensional vector s onto the range R ( V ) of a matrix V is the unique vector ˆ s in R ( V ) with the property that s ˆ s is orthogonal to every column of V . It is the point in R ( V ) closest to s . • ˆ s is obtained by solving the n × n system of equations ( V T V ) c = V T s for c , followed by forming the linear combination ˆ s = Vc . Theory and Examples: 1 . The inner product of two mdimensional realvalued vectors is the same as their dot product: h a , b i = m X i =1 a i b i It is symmetric, i.e.,It is symmetric, i....
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 Spring '08
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 Linear Algebra, Vectors, Vector Space

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