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In particular, you must clearly indicate the positive direction of each loop you are considering, and ensure that the voltage drop across every resistor and electrical source on the loop[r]
(1)c
W W L Chen, 1982, 2008
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
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Chapter 1
LINEAR EQUATIONS
1.1 Introduction
Example 1.1.1 Try to draw the two lines
3x + 2y = 5, 6x + 4y =
It is easy to see that the two lines are parallel and not intersect, so that this system of two linear equations has no solution
Example 1.1.2 Try to draw the two lines
3x + 2y = 5, x + y =
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
3x + 2y = 5, 6x + 4y = 10
(2)In these three examples, we have shown that a system of two linear equations on the plane R2 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 a11x1+ a12x2+ + a1nxn= b1,
a21x1+ a22x2+ + a2nxn= b2,
am1x1+ am2x2+ + amnxn= bm,
(1)
with n variables x1, x2, , xn 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
a11 a12 a1n
a21 a22 a2n
am1 am2 amn
b1
b2
bm
(2)
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 Ax = b, where
A =
a11 a12 a1n
a21 a22 a2n
am1 am2 amn
and b=
b1
b2
bm
represent the coefficients and
x=
x1
x2
xn
represents the variables
Example 1.1.4 The array
1 10 1 1
4
Then this represents the system of equations
x2+ x3+ 2x4+ x5= 4,
x1+ 3x2+ x3+ 5x4+ x5= 5,
2x1+ 4x2 + 7x4+ x5= 3,
(5) essentially the same as the system (4), the only difference being that the first and second equations have been interchanged Any solution of the system (4) is a solution of the system (5), and vice versa Example 1.2.2 Consider the array (3) Let us add times the second row to the first row to obtain
1 30 1 1
13
Then this represents the system of equations
x1+ 5x2+ 3x3+ 9x4+ 3x5= 13,
x2+ x3+ 2x4+ x5= 4,
2x1+ 4x2 + 7x4+ x5= 3,
(6) essentially the same as the system (4), the only difference being that we have added times the second equation to the first equation Any solution of the system (4) is a solution of the system (6), and vice versa
Example 1.2.3 Consider the array (3) Let us multiply the second row by to obtain
1 10 2 2