AppendixB

AppendixB - John Riley 15 August 2011 APPENDIX B: MAPPINGS...

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© John Riley 15 August 2011 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 9 Functions of 2 variables Partial and total derivatives Functions of n variables Contour Sets Concave and quasi-concave functions E x e r c i s e s B.3 TRANSFORMATIONS OF VECTORS 3 0 M a t r i x Q u a d r a t i c F o r m Quadratic Approximation of a function I n v e r s e m a t r i x C r a m e r s R u l e Exercises B.4 SYSTEMS OF LINEAR DIFFERENCE EQUATIONS 45 57 pages 27 figures
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Appendix B page 1 B .1 VECTORS AND SETS Key ideas: orthogonal vector, hyperplanes, 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 xx = 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 yx > . (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 (,) dyz between any two vectors 1 ( ,. .., ) n y yy = and 1 ( ,. .., ) n zz z = . If 1 n = , a natural measure is the absolute value of the difference so that (,) y z =− . Similarly with 2 n = , a natural measure is the Euclidean distance yz between the two vectors. Appealing to Pythagoras Theorem, 2 22 11 2 2 () ( ) −=− +− . Extending this to n -dimensions, the square of Euclidean distance between ordered n -tuples is defined as follows: 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 jj j yz 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 ii i = ⋅= From this definition, it follows that the distributive law for multiplying vectors is the same as for numbers. Distributive law for vector multiplication abc abac ⋅+=⋅+ 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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Appendix B page 3 The length of each vector is its distance to the origin. If x and p are perpendicular it follows from Pythagoras Theorem that the square of the hypotenuse equals the sum of the squares of the other two sides, that is, 22 2 p xp x −= + . (B.4-1) From the definition of a vector product, the square of Euclidean distance between two vectors can then be written as follows.
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AppendixB - John Riley 15 August 2011 APPENDIX B: MAPPINGS...

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