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l_notes30 - Lecture XXX nVectors and Matrices 1 nVectors We...

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1 Lecture XXX n-Vectors and Matrices n-Vectors We define E n to be the n -dimensional Euclidean space, and R n to be the set of points in E n . Hence R n is the set of all ordered n -tuples of real numbers ( x 1 , . . . , x n ). Ordered n -tuples allow repetitions, i.e. x i = x j is possible for i = j , but if x i = x j , then ( x 1 , . . . , x i , . . . , x j , . . . , x n ) = ( x 1 , . . . , x j , . . . , x i , . . . , x n ). These n -tuples can be viewed as vectors in the n -dimensional Euclidean space. Hence we will call them n-vectors . Definition 1 Let P = ( p 1 , p 2 , . . . , p n ) and Q ( q 1 , q 2 , . . . , q n ) . We say that the distance between P and Q is d ( P, Q ) = ( p 1 q 1 ) 2 + ( p 2 q 2 ) 2 + · · · + ( p n q n ) 2 . We define 0 = (0 , 0 , . . . , 0). On n -vectors, we can define operations analogous to the ones on 3-vectors: 1. Multiplication of a vector by a scalar A = ( a 1 , a 2 , . . . , a n ) and let k be a scalar. Then k Let A = ( ka 1 , ka 2 , . . . , ka n ). 2. Vector addition A + Let A = ( a 1 , a 2 , . . . , a n ) and B = ( b 1 , b 2 , . . . , b n ). Then B = ( a 1 + b 1 , a 2 + b 2 , . . . , a n + b n ). 3. Dot product A · Let A = ( a 1 , a 2 , . . . , a n ) and B = ( b 1 , b 2 , . . . , b n ). Then B = a 1 b 1 + a 2 b 2 + + a n b n . · · · 4. Magnitude of
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