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09Appendix-B-1

09Appendix-B-1 - John Riley 2 September 2009 APPENDIX B...

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© John Riley 2 September 2009 APPENDIX B: MAPPINGS OF VECTORS B.1 VECTORS AND SETS 1 Orthogonal vectors and hyper-planes Convex sets Open and Closed Sets B.2 FUNCTIONS OF VECTORS 8 Functions of 2 variables Partial and total derivatives Functions of n variables Contour Sets Concave and quasi-concave functions Exercises B.3 TRANSFORMATIONS OF VECTORS 29 Matrix Quadratic Form Quadratic Approximation of a function Inverse matrix Cramer’s Rule Exercises B.4 SYSTEMS OF LINEAR DIFFERENCE EQUATIONS 43 54 pages
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Appendix B page 1 B .1 VECTORS AND SETS Key ideas: orthogonal vector, hyper-planes, convex sets, open and closed sets We now extend our analysis to ordered n -tuples or “vectors.” Each component of the vector 1 ( ,..., ) n x x x = is a real number. Where it is important to make clear the dimension of the vector, we write n x \ . If each component is positive, then x is positive and we write n x + \ . We will describe the vector y as being larger than the vector x if every component of y is at least as large as x and write y x . If at least one component of y is strictly larger we will say that y is strictly larger than x and write y x > . (We will occasionally use the notation x y ± to indicate that all components are strictly larger.) A neighborhood of a vector is the set of points near it. Thus to define a neighborhood we need some measure of distance ( , ) d y z between any two vectors 1 ( ,..., ) n y y y = and 1 ( ,..., ) n z z z = . If 1 n = , a natural measure is the absolute value of the difference so that ( , ) d y z y z = . Similarly, with 2 n = , a natural measure is the Euclidean distance y z between the two vectors. Appealing to Pythagoras Theorem, 2 2 2 1 1 2 2 ( ) ( ) y z y z y z = + . Extending this to n -dimensions, the square of Euclidean distance between ordered n -tuples is defined as follows. 2 2 y z 1 1 y z y z 1 x 2 x Fig. B.1-1: Distance between 2 vectors y z
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Appendix B page 2 2 2 1 ( ) n j j j y z y z = = It is sometimes more convenient to express distance as a vector product. Vector Product The product (“inner product” or “sumproduct”) of two n -dimensional vectors y and z is. 1 n i i i y z y z = = From this definition, it follows that the distributive law for multiplying vectors is the same as for numbers. Distributive law for vector multiplication ( ) a b c a b a c + = + Orthogonal Vectors Two vectors, x and p are depicted below in 3-dimensional space. Fig. B.1-2: Orthogonal Vectors 1 x O x 2 x 3 x x p p p x
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