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Unformatted text preview: 732 8.1. FUNCTIONS OF TWO VARIABLES 8.1 Functions of Two Variables 8.2 Partial Derivatives, XX 8.3 Optimization XX 8.4 Linear Equations and Matrices in Two Dimensions In Chapters XX we represented points in the x- y plane as ordered pairs ( x,y ). Another way of referring to points in the plane is to represent them as a 2-dimensional vectors z written as columns with two entries x and y in the first and second rows respectively; i.e in the form z = x y . Note that the variables x and y are written italics as we have done through out this book for variables that have a single value. The variable z , however, is written in boldface type to remind us that it has more than one value: in this case two values—that of its first element x and that of its second element y . We also note that any point c = a b in the plane defines a line through the origin that has slope a b (Fig. 1.2) and, hence, the equation of this line is y = a b x . PLACE FIG HERE Figure 8.2: The line through the origin that also passes through the point defined by the vector c with first and second row elements a and b has the equation y = a b x In the algebra of numbers that are points on the real line, if we take any number x 1 on this line and multiply it by a constant a , we get a new number x 2 = ax 1 on this line. In our algebra of vector points in the plane we want an object, which we shall call a matrix A such that formally if we “multiply” a vector z 1 by a matrix A , we get a new vector z 2 represented by the equation z 2 = A z 1 . What should the matrix A look like and how should the implied multiplication operation Az 1 work to obtain an arithmetic and algebra of 2-dimensional vectors that under the subsumes our familiar arithmetic and algebra of points on a line as a special case. This problem was solved by 18 th and 19 th Century mathematicians in their development of a general theory of linear transformations and matrix algebra. Here we confine ourselves to the bare minimum of what is necessary to be able to find solutions to algebraic and differential equations involving two variables x and y . Before we can begin to explore the algebra of vectors we need some definitions. Vector Equality Two vectors c = a b and r = p q are equal if and only if a = p and b = q . © 2008 Schreiber, Smith & Getz 8.4. LINEAR EQUATIONS AND MATRICES IN TWO DIMENSIONS 733 Vector Addition If c = a b and r = p q then c + r = a b + p q = a + p b + q . Although vectors are written in column form, we can write them in row form as well. We we write a column vector as a row rather than a column, we call the row form the transpose and not this fact by placing a T (read “transpose”) after the vector. Thus the vector z = x y can also be written as z = ( x,y ) T . Multiplication of two 2-D vectors from our narrow point of view in this chapter can only be performed if the first vector is in its transpose form....
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This note was uploaded on 02/09/2011 for the course EEP 115 taught by Professor Waynem.getz during the Fall '10 term at Berkeley.
- Fall '10