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Unformatted text preview: 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). 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 )....
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