SYSTEMS OF EQUATIONS
1. S YSTEMS OF EQUATIONS
A system of equations is a nite set of equations containing more than one variable. We sometimes
refer to a system of equations as a set of simultaneous equations in the sense that we seek values for
the varia
CONVEXITY AND OPTIMIZATION
1. C ONVEX SETS 1.1. Definition of a convex set. A set S in Rn is said to be convex if for each x1 , x2 S, the line segment x1 + (1-)x2 for (0,1) belongs to S. This says that all points on a line connecting two points in the set
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
1. Systems of Equations and Matrices
1.1. Representation of a linear system. The general system of m equations in n unknowns can
be written
a11 x1
+
a12 x2
+
+
a1n xn
=
b1
a21 x1
+
a22 x2
+
+
a2n xn
=
b2
a31 x1
+
FUNCTIONS AND EQUATIONS
1. S ETS AND
SUBSETS
1.1. Denition of a set. A set is any collection of objects which are called its elements. If x is an element
of the set S, we say that x belongs to S and write
x S.
If y does not belong to S, we write
y S.
The
INTRODUCTION TO MATRIX ALGEBRA
1. D EFINITION
OF A MATRIX AND A VECTOR
1.1. Denition of a matrix. A matrix is a rectangular array of numbers arranged into rows and
columns. It is written as
a11 a12 . . . a1n
a21 a22 . . . a2n
.
.
.
.
(1)
.
.
.
.
.
REVIEW OF SIMPLE UNIVARIATE CALCULUS
1. A PPROXIMATING
CURVES WITH LINES
1.1. The equation for a line. A linear function of a real variable x is given by
y = f ( x ) = ax + b,
a and b are constants
(1)
The graph of linear equation is a straight line. The
SINGLE VARIABLE OPTIMIZATION
1. D EFINITION
OF LOCAL MAXIMA AND LOCAL MINIMA
1.1. Note on open and closed intervals. 1.1.1. Open interval. If a and b are two numbers with a < b, then the open interval from a to b is the collection of all numbers which are
SIMPLE MULTIVARIATE CALCULUS
1. R EAL - VALUED F UNCTIONS OF S EVERAL VARIABLES
1.1. Denition of a real-valued function of several variables. Suppose D is a set of n-tuples of real numbers (x1 , x2 , x3 , . . . , xn ). A real-valued function f on D is a r
SIMPLE MULTIVARIATE OPTIMIZATION
1. D EFINITION OF LOCAL MAXIMA AND LOCAL MINIMA 1.1. Functions of 2 variables. Let f(x1 , x2 ) be defined on a region D in
2
containing the point (a, b). Then
a: f(a, b) is a local maximum value of f if f(a, b) f(x1 , x2 )
SIMPLE CONSTRAINED OPTIMIZATION
1. I NTUITIVE I NTRODUCTION TO C ONSTRAINED O PTIMIZATION
Consider the following function which has a maximum at the origin.
y = f (x1 , x2 ) = 49 x2 x2
1
2
The graph is contained in gure 1.
(1)
F IGURE 1. The function y =
University of Toronto Mississauga, Erindale College
Eco220Y Quantitative Methods Fall 2012
G.J.Anderson
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Erindale Office (905) 828 3904
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