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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 184.108.40.206.4 Key Points: The norm, or length, of a real-valued vector a is given by k a k = h a , a i 1 / 2 , where h , i denotes inner (dot) product. The angle between two vectors a and b satisfies cos = h a , b i k a k k b k The (orthogonal) projection of b on a is the vector a such that b- a is orthogonal to a , i.e., h b- a , a i = 0. Therefore = 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 m-dimensional vector s onto the range R ( V ) of a matrix V is the unique vector s in R ( V ) with the property that s- s is orthogonal to every column of V . It is the point in R ( V ) closest to s . 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 s = Vc . Theory and Examples: 1 . The inner product of two m-dimensional real-valued 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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This note was uploaded on 10/18/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.
- Spring '08