c1s2 - 1 1.2 The Norm and Dot Product Chapter 1. Vectors,...

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1.2 The Norm and Dot Product 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.2. The Norm and Dot Product Defnition 1.5. Let ~v =[ v 1 ,v 2 ,...,v n ] R n . The norm or magnitude of is k k = q v 2 1 + v 2 2 + ··· + v 2 n = v u u t n X l =1 ( v l ) 2 . Theorem 1.2. Properties oF the Norm in R n . For all ~v, ~w R n and for all scalars r R ,wehave : 1. k k≥ 0and k k =0ifandon lyif = ~ 0 . 2. k r~v k = | r |k k . 3. k + ~w k≤k k + k k (the Triangle Inequality). Note. 1 and 2 are easy to see and we will prove 3 later in this section.
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1.2 The Norm and Dot Product 2 Note. A picture for the Triangle Inequality is: 1.2.22, page 22 Defnition. A vector with norm 1 is called a unit vector . When writing, unit vectors are frequently denoted with a “hat”: ˆ i. Example. Page 31 number 8. Defnition 1.6. The dot product for ~v =[ v 1 ,v 2 ,...,v n ]and ~w = [ w 1 ,w 2 ,...,w n ]is · = v 1 w 1 + v 2 w 2 + ··· + v n w n = n X l =1 v l w l .
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1.2 The Norm and Dot Product 3 Notice. If we let θ be the angle between nonzero vectors ~v and ~w ,then we get by the Law of Cosines: 1.2.24, page 23 k k 2 + k k 2 = k k +2 k k k cos
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c1s2 - 1 1.2 The Norm and Dot Product Chapter 1. Vectors,...

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