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la01-le - LINEAR ALGEBRA W W L CHEN c W W L Chen 1982 2005...

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LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2005. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 1 LINEAR EQUATIONS 1.1. Introduction Example 1.1.1. Try to draw the two lines 3 x + 2 y = 5 , 6 x + 4 y = 5 . It is easy to see that the two lines are parallel and do not intersect, so that this system of two linear equations has no solution. Example 1.1.2. Try to draw the two lines 3 x + 2 y = 5 , x + y = 2 . It is easy to see that the two lines are not parallel and intersect at the point (1 , 1), so that this system of two linear equations has exactly one solution. Example 1.1.3. Try to draw the two lines 3 x + 2 y = 5 , 6 x + 4 y = 10 . It is easy to see that the two lines overlap completely, so that this system of two linear equations has infinitely many solutions. Chapter 1 : Linear Equations page 1 of 21
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Linear Algebra c W W L Chen, 1982, 2005 In these three examples, we have shown that a system of two linear equations on the plane R 2 may have no solution, one solution or infinitely many solutions. A natural question to ask is whether there can be any other conclusion. Well, we can see geometrically that two lines cannot intersect at more than one point without overlapping completely. Hence there can be no other conclusion. In general, we shall study a system of m linear equations of the form (1) a 11 x 1 + a 12 x 2 + . . . + a 1 n x n = b 1 , a 21 x 1 + a 22 x 2 + . . . + a 2 n x n = b 2 , . . . a m 1 x 1 + a m 2 x 2 + . . . + a mn x n = b m , with n variables x 1 , x 2 , . . . , x n . Here we may not be so lucky as to be able to see geometrically what is going on. We therefore need to study the problem from a more algebraic viewpoint. In this chapter, we shall confine ourselves to the simpler aspects of the problem. In Chapter 6, we shall study the problem again from the viewpoint of vector spaces. If we omit reference to the variables, then system (1) can be represented by the array (2) a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . a m 1 a m 2 . . . a mn b 1 b 2 . . . b m of all the coefficients. This is known as the augmented matrix of the system. Here the first row of the array represents the first linear equation, and so on. We also write A x = b , where A = a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . a m 1 a m 2 . . . a mn and b = b 1 b 2 . . . b m represent the coefficients and x = x 1 x 2 . . . x n represents the variables.
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