Chapter 5 - CHAPTER 5 Linear Algebra 5.1 Introduction...

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CHAPTER 5 Linear Algebra 5.1 Introduction Linear algebra is an important mathematical subject independent of its connection to differential equations. However, it is the relation between differential equations and linear algebra that is important to us and so we will use this relationship to motivate our study of linear algebra. In Chapter 3 we studied second order linear differential equations: y ±± + p ( x ) y ± + q ( x ) y = f ( x ) (1) where p, q and f are continuous functions on an interval I . The focus of our study was on the homogeneous equation y ±± + p ( x ) y ± + q ( x ) y =0 , (2) the so-called reduced equation of equation (1). We saw that the general solution of (2) is given by y = C 1 y 1 ( x )+ C 2 y 2 ( x ) where C 1 and C 2 are arbitrary constants and y 1 and y 2 are linearly independent solutions of the equation. We also saw that an initial-value problem y ±± + p ( x ) y ± + q ( x ) y ; y ( a )= α, y ± ( a β required us to solve the system of equations y 1 ( a ) C 1 + y 2 ( a ) C 2 = α y ± 1 ( a ) C 1 + y ± 2 ( a ) C 2 = β In the next chapter we will consider higher-order linear differential equations as a lead in to our study of systems of linear differential equations. An n th -order linear differential equation is an equation of the form y ( n ) + p n - 1 ( x ) y ( n - 1) + p n - 2 ( x ) y ( n - 2) + ··· + p 1 ( x ) y ± + p 0 ( x ) y = f ( x ) (3) where p 0 ,p 1 ,. . . n - 1 and f are continuous functions on some interval I . As in the case of second order equations, the focus is on homogeneous equations: y ( n ) + p n - 1 ( x ) y ( n - 1) + p n - 2 ( x ) y ( n - 2) + + p 1 ( x ) y ± + p 0 ( x ) y . (4) All of the theorems in Chapter 3 carry over verbatim to n th -order equations. In particular, to obtain the general solution of (4) we need to Fnd n linearly independent solutions y 1 ,y 2 . . n of (4). The general solution is then given by y = C 1 y 1 ( x C 2 y 2 ( x + C n y n ( x ) . This is the Frst major question. We know what it means for two functions to be linearly independent (one is not a multiple of the other), but what does it mean for n functions to be linearly independent? And, given a set of n functions, how can we determine whether or not they are linearly independent? 155
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To solve the n th -order initial-value problem y ( n ) + p n - 1 ( x ) y ( n - 1) + p n - 2 ( x ) y ( n - 2) + ··· + p 1 ( x ) y ± + p 0 ( x ) y =0 ; y ( a )= α 0 ,y ± ( a α 1 ,. . . ( n - 1) ( a α n - 1 we would need to solve the system of n equations in n unknowns y 1 ( a ) C 1 + y 2 ( a ) C 2 + + y n ( a ) C n = α 0 y ± 1 ( a ) C 1 + y ± 2 ( a ) C 2 + + y ± n ( a ) C n = α 1 . . . y ( n - 1) 1 ( a ) C 1 + y ( n - 1) 2 ( a ) C 2 + + y ( n - 1) n ( a ) C n = α n - 1 . This is the second major question. While it is easy to solve two equations in two unknowns, how do we solve “large” systems consisting of m equations in n unknowns? The primary purpose of this chapter is to address these two questions along with a number of other topics that are closely related to these questions. 156
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5.2 Systems of Linear Equations, Some Geometry A linear (algebraic) equation in n unknowns, x 1 ,x 2 ,.
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Chapter 5 - CHAPTER 5 Linear Algebra 5.1 Introduction...

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