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Unformatted text preview: Vector Spaces; n-Dimensional Euclidean Space Introduction The Space E n consists of all ordered n-tuples of real or complex numbers X = x 1 x 2 . . . x n , Y = y 1 y 2 . . . y n , V = x y z (in E 3 ) , etc ., where x i , y i , are scalars (real or complex numbers), called the components of X and Y , respectively. The qualification ordered means that, e.g., 1 2 3 6 = 2 1 3 . In many physical applications one wishes to restrict attention to vectors having only real components; in such cases it is common to refer to the space as R n . For the most part we will consider the more general case of E n ; when we give a definition referring to vectors in E n it will be understood that the obvious restriction of that definition to R n applies as well, unless we explicitly state to the contrary. Vectors in E n can be thought of either as points in space or as directed magnitudes , the latter often represented graphically by arrows in the special cases of R 2 and R 3 . Quantities representable by a single number (e.g., speed, time, weight, density) are scalar quantities. In particular, real and complex numbers are scalars in these notes. Real numbers are quantities which have only magnitude , | | , and sign (+ or -). Complex numbers are really two dimensional vectors with a particular algebraic structure; the sign is replaced by the argument of a complex number, essentially the angular component of the polar representation of the vector. 1 Magnitude of a Vector If X = ( x 1 , x 2 , ..., x n ) is an n-dimensional vector, its magnitude or norm is k X k = q | x 1 | 2 + | x 2 | 2 + + | x n | 2 . This is the length of a vector in the ordinary, Euclidean , sense. Thus - 2 1 3 = q (- 2) 2 + (1) 2 + (3) 2 = 14 = 2 7 . 1 + i i 2- i = q (1 2 + 1 2 ) + (0 2 + 1 2 ) + (2 2 + (- 1) 2 ) = 8 = 2 2 . Algebraic Operations on Vectors The operations of addition and subtraction of vectors are performed com- ponentwise ; the result is another vector of the same size or dimension : Z = X Y : z 1 z 2 . . . z n = x 1 x 2 . . . x n y 1 y 2 . . . y n , i . e .,z k = x k y k , k = 1 , 2 , ..., n. Thus, for example, 2 1 + - 3- 1 1 = 2- 3- 1 1 + 1 = - 1- 1 2 ; 1 + i i 2- i + - 3- 1 1 + 2 i = - 2 + i- 1 + i 3 + i ....
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