cal5 - Chapter Five More Dimensions 5.1 The Space R n We...

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5.1 Chapter Five More Dimensions 5.1 The Space R n We are now prepared to move on to spaces of dimension greater than three. These spaces are a straightforward generalization of our Euclidean space of three dimensions. Let n be a positive integer. The n-dimensional Euclidean space R n is simply the set of all ordered n- tuples of real numbers x = ( , , , ) x x x n 1 2 K . Thus R 1 is simply the real numbers, R 2 is the plane, and R 3 is Euclidean three-space. These ordered n- tuples are called points , or vectors . This definition does not contradict our previous definition of a vector in case n =3 in that we identified each vector with an ordered triple ( , , ) x x x 1 2 3 and spoke of the triple as being a vector. We now define various arithmetic operations on R n in the obvious way. If we have vectors x = ( , , , ) x x x n 1 2 K and y = ( , , , ) y y y n 1 2 K in R n , the sum x y + is defined by x y + = + + + ( , , , ) x y x y x y n n 1 1 2 2 K , and multiplication of the vector x by a scalar a is defined by a ax ax ax n x = ( , , , ) 1 2 K . It is easy to verify that a a a ( ) x y x y + = + and ( ) a b a b + = + x x x . Again we see that these definitions are entirely consistent with what we have done in dimensions 1, 2, and 3-there is nothing to unlearn. Continuing, we define the length , or norm of a vector x in the obvious manner | | x = + + + x x x n 1 2 2 2 2 K . The scalar product of x and y is
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5.2 x y = + + + = = x y x y x y x y n n i i i n 1 1 2 2 1 K .
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This note was uploaded on 10/27/2009 for the course MBA 1918272 taught by Professor Peter during the Fall '09 term at Aberystwyth University.

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cal5 - Chapter Five More Dimensions 5.1 The Space R n We...

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