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Unformatted text preview: University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A23 Winter 2008 Lecture 1 Sophie Chrysostomou Definition : If n is a positive integer, the Euclidean n-space, R n , is the collection of all ordered n-tuples of real numbers. There are two types of n-tuples in R n : ( a 1 , a 2 , , a n ) [ a 1 , a 2 , , a n ] The zero vector in R n is the vector containing zeroes in all of its compo- nents: = [0 , , , 0]. Geometric Interpretation of Vectors. In R In R 2 : In R 3 1 We can generalize and say that the vector a = [ a 1 , a 2 , , a n ], in its standard position, is the arrow that starts at the origin (0 , , , 0) and ends at the point ( a 1 , a 2 , , a n ) . A vector of the same length and direction as a , is the vector a translated to another position in R n . For this reason it is also called a . Vector Algebra in R n : Let v = [ v 1 , v 2 , , v n ] and w = [ w 1 , w 2 , , w n ] be in...
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