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Unformatted text preview: Euclidean space 1 Chapter 1 Euclidean space A. The basic vector space We shall denote by R the field of real numbers. Then we shall use the Cartesian product R n = R × R × ... × R of ordered ntuples of real numbers ( n factors). Typical notation for x ∈ R n will be x = ( x 1 ,x 2 ,...,x n ) . Here x is called a point or a vector , and x 1 , x 2 ,...,x n are called the coordinates of x . The natural number n is called the dimension of the space. Often when speaking about R n and its vectors, real numbers are called scalars . Special notations: R 1 x R 2 x = ( x 1 ,x 2 ) or p = ( x,y ) R 3 x = ( x 1 ,x 2 ,x 3 ) or p = ( x,y,z ) . We like to draw pictures when n = 1, 2, 3; e.g. the point ( 1 , 3 , 2) might be depicted as 2 Chapter 1 We define algebraic operations as follows: for x , y ∈ R n and a ∈ R , x + y = ( x 1 + y 1 ,x 2 + y 2 ,...,x n + y n ); ax = ( ax 1 ,ax 2 ,...,ax n ); x = ( 1) x = ( x 1 , x 2 ,..., x n ); x y = x + ( y ) = ( x 1 y 1 ,x 2 y 2 ,...,x n y n ) . We also define the origin (a/k/a the point zero ) 0 = (0 , ,..., 0) . (Notice that 0 on the left side is a vector, though we use the same notation as for the scalar 0.) Then we have the easy facts: x + y = y + x ; ( x + y ) + z = x + ( y + z ); 0 + x = x ; in other words all the x x = 0; “usual” algebraic rules 1 x = x ; are valid if they make ( ab ) x = a ( bx ); sense a ( x + y ) = ax + ay ; ( a + b ) x = ax + bx ; x = 0; a 0 = 0 . Schematic pictures can be very helpful. One nice example is concerned with the line determined by x and y (distinct points in R n ). This line by definition is the set of all points of the form (1 t ) x + ty, where t ∈ R . Here’s the picture: Euclidean space 3 This picture really is more than just schematic, as the line is basically a 1dimensional object, even though it is located as a subset of ndimensional space. In addition, the closed line segment with end points x and y consists of all points as above, but with 0 ≤ t ≤ 1. This segment is shown above in heavier ink. We denote this segment by [ x,y ]. We now see right away the wonderful interplay between algebra and geometry , something that will occur frequently in this book. Namely, the points on the above line can be described completely in terms of the algebraic formula given for the line. On the other hand, the line is of course a geometric object. It is very important to get comfortable with this sort of interplay. For instance, if we happen to be discussing points in R 5 , we probably have very little in our background that gives us geometric insight to the nature of R 5 . However, the algebra for a line in R 5 is very simple, and the geometry of a line is just like the geometry of R 1 ....
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This note was uploaded on 09/28/2008 for the course MATH 1220 taught by Professor Hatcher during the Fall '08 term at Cornell University (Engineering School).
 Fall '08
 Hatcher
 Calculus, Real Numbers, Factors, Vector Space

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