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Unformatted text preview: UNIT 5 VECTOR SPACES INTRO: This Unit is on vector spaces and on the dot product of a pair of vectors in the vector spaces R 2 and R 3 . First, though, some preliminary remarks. 1. Abstract Vector Spaces Nearly all students will have already been exposed to the notion of a vector, perhaps defined as a quantity with magnitude and direction, perhaps defined as a pair of numbers ( a,b ) or a triple of numbers ( a,b,c ) with rules for adding and multiplying by constants. These are called the vector spaces R 2 and R 3 . Because these two vector spaces can be closely idenified with physical two dimensional and three dimensional space, we are comfortable saying that these vectors have magnitude (length) and direction. Actually, the definition of a vector is much more encompassing. To begin it is necessary to define the entire set of vectors V , called the vector space, and to define how to add and to multiply by constants the elements of the set. Any element in the set V defined as the vector space is called a vector ~v V . The small arrow above the vector v is unnecessary; it is used simply as a device to remind us that ~v is a vector. In general, a vector space may have no concept of direction and no concept of magnitude. For that reason, defining a vector as an object having magnitude and direction would be misleading. For example, in many technology areas complex systems are described using a vector space of functions, in which the functions will be the vectors and there may be no connection with a notion of direction. We, however, are going to be working exclusively with the two vector spaces R 2 and R 3 , so there will be no drawback to thinking of vectors as objects with magnitude and direction, at least in this course. For further information on vectors in more general settings, please refer to the Appendix of this Unit. 2. Vectors in R 2 and R 3 In many elementary courses in linear algebra, vectors are written either as a row ( a 1 ,a 2 ,...,a n ) or as a column a 1 a 2 a n of n real numbers, and the set of all such vectors of fixed length n is called the vector space R n . We shall be interested only in the cases n = 2 and n = 3, ie, only in R 2 and R 3 . Editors VECTOR GEOMETRY 59 GREENBERG (and we) prefer to have such vectors written as rows in textbook explanations, since it takes up substantially less room in printing a paragraph, although if the vectors are to be multiplied in the normal way by a matrix, then it is important to write them as columns. Since we will not be carrying out matrix multiplication, we will generally display our vectors as rows. Except for operations with matrices, the distinction between row vector and column vector is of no importance. For additional clarity, many textbooks surround the vector with brackets < > and, if given a letter name, place an arrow above. So, ~v = < a,b > is in the vector space R 2 , while ~v = < a,b,c > is in the vector space R 3 . This enables the author to distinguish between a....
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