Author: haroonkhan
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Operations on vectors
We are quite used to elementary operations on numbers, such as addition, multiplication, and division. Not all of them can be sensibly extended to the vector case. Still, we will find some of them quite useful, namely: Vector addition Addition is defined for pairs of vectors having the same dimension. If , we define: For instance Since…
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VECTOR ALGEBRA
Vectors are an intuitive concept that we get acquainted with in highschool mathematics. In ordinary two- and three-dimensional geometry, we deal with points on the plane or in the space. Such points are associated with coordinates in a Cartesian reference system. Coordinates may be depicted as vectors, as shown in Fig. 3.4; in physics, vectors are…
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Cramer’s rule
As a last approach, we consider Cramer’s rule, which is a handy way to solve systems of two or three equations. The theory behind it requires more advanced concepts, such as matrices and their determinants, which are introduced below. We anticipate here a few concepts so that readers not interested in advanced multivariate statistics can…
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Gaussian elimination
Gaussian elimination, with some improvements, is the basis of most numerical routines to solve systems of linear equations. Its rationale is that the following system is easy to solve: Such a system is said to be in upper triangular form, as nonzero coefficients form a triangle in the upper part of the layout. A system in…
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Substitution of variables
A basic (highschool) approach to solving a system of linear equations is substitution of variables. The idea is best illustrated by a simple example. Consider the following system: Rearranging the first equation, we may express the first variable as a function of the second one: and plug this expression into the second equation: Then we…
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SOLVING SYSTEMS OF LINEAR EQUATIONS
The theory, as well as the computational practice, of solving systems of linear equations is relevant in a huge list of real-life settings. In this section we just outline the basic solution methods, without paying due attention to bothering issues such as the existence and uniqueness of a solution, or numerical precision. A linear equation…
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A MOTIVATING EXAMPLE: BINOMIAL OPTION PRICING
Options are financial derivatives that have gained importance, as well as a bad reputation, over the years. In Section 1.3.1 we considered forward contracts, another type of derivative. With a forward contract, two parties agree on exchanging an asset or a commodity, called the underlying asset, at a prescribed time in the future, for a fixed price determined…
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A business view of definite integral
In this section we try to further motivate the use of definite integral, at least conceptually, for business management problems. To do so, we use the EOQ model of Section 2.1 once again. There, we have claimed that the contribution of inventory holding cost to average total cost per unit time is In the reasoning, a key…
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Improper integrals
So far, we have considered the integral of a continuous function on a bounded interval. The idea can be generalized to unbounded intervals and to functions featuring certain types of discontinuity. In fact, the integral might not exist, because the function has pathological behavior; in other cases, it could go to infinity, which may well…
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Calculating definite integrals
Using the definition above to compute an integral is cumbersome, to say the least. It may work in some simple cases, but we certainly need something more handy. Luckily, the following theorem, which really deserves the name fundamental,27 provides us with a practical way to compute definite integrals. THEOREM 2.22 (Fundamental theorem of calculus) Let F(x) be a function…