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]
Trang 1W W L CHEN
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.
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It is easy to see that the two lines are parallel and do not intersect, so that this system of two linearequations has no solution
Example 1.1.2 Try to draw the two lines
3x + 2y = 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
3x + 2y = 5,6x + 4y = 10
It is easy to see that the two lines overlap completely, so that this system of two linear equations hasinfinitely many solutions
Trang 2In these three examples, we have shown that a system of two linear equations on the plane R2 mayhave no solution, one solution or infinitely many solutions A natural question to ask is whether therecan be any other conclusion Well, we can see geometrically that two lines cannot intersect at more thanone 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
If we omit reference to the variables, then system (1) can be represented by the array
am1 am2 amn
b1
b2
represents the variables
Example 1.1.4 The array
1 3 1 5 10 1 1 2 1
2 4 0 7 1
453
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 havebeen 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 2 times the second row to the first row to obtain
1 5 3 9 30 1 1 2 1
2 4 0 7 1
1343
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 2 times the secondequation to the first equation Any solution of the system (4) is a solution of the system (6), and viceversa
Example 1.2.3 Consider the array (3) Let us multiply the second row by 2 to obtain
1 3 1 5 10 2 2 4 2
2 4 0 7 1