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Unformatted text preview: 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
3x + 2y = 5,
6x + 4y = 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
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 has
inﬁnitely many solutions.
Chapter 1 : Linear Equations page 1 of 21 c Linear Algebra W W L Chen, 1982, 2005 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
a11 x1 + a12 x2 + . . . + a1n xn = b1 ,
a21 x1 + a22 x2 + . . . + a2n xn = b2 ,
.
.
.
am1 x1 + am2 x2 + . . . + amn xn = 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 a21
.
.
. ...
... am1 (2) a12
a22
.
.
.
am2 . . . amn a1n
a2n
.
.
. b1
b2 .
.
.
bm 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 a11 a21
A= .
.
. a12
a22
.
.
. ...
... a1n
a2n .
.
. am1 am2 ... amn and b1 b2 b= . .
.
bm represent the coeﬃcients and x1 x2 x= . .
.
xn
represents the variables.
Example 1.1.4. The array (3) 1
0
2 3
1
4 1
1
0 5
2
7 1
1
1 5
4
3 represents the system of 3 linear equations (4) x1 + 3x2 + x3 + 5x4 + x5 = 5,
x2 + x3 + 2x4 + x5 = 4,
2x1 + 4x2 Chapter 1 : Linear Equations + 7x4 + x5 = 3,
page 2 of 21 c Linear Algebra W W L Chen, 1982, 2005 with 5 variables x1 , x2 , x3 , x4 , x5 . We can also write 1
0
2 3
1
4 1
1
0 5
2
7 x1 1 x2 5 1 x3 = 4 . 1
x4
3
x5 1.2. Elementary Row Operations
Let us continue with Example 1.1.4.
Example 1.2.1. Consider the array (3). Let us interchange the ﬁrst and second rows to obtain 011214
1 3 1 5 1 5.
240713
Then this represents the system of equations
x2 + x3 + 2x4 + x5 = 4,
(5) x1 + 3x2 + x3 + 5x4 + x5 = 5,
2x1 + 4x2
+ 7x4 + x5 = 3, 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 2 times the second row to the ﬁrst row to obtain
1 5 3 9 3 13
0 1 1 2 1
4 .
3
24071
Then this represents the system of equations (6) x1 + 5x2 + 3x3 + 9x4 + 3x5 = 13,
x2 + x3 + 2x4 + x5 = 4,
2x1 + 4x2 + 7x4 + x5 = 3, essentially the same as the system (4), the only diﬀerence being that we have added 2 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 2 to obtain 131515
0 2 2 4 2 8.
240713
Then this represents the system of equations (7) Chapter 1 : Linear Equations x1 + 3x2 + x3 + 5x4 + x5 = 5,
2x2 + 2x3 + 4x4 + 2x5 = 8,
2x1 + 4x2
+ 7x4 + x5 = 3,
page 3 of 21 c Linear Algebra W W L Chen, 1982, 2005 essentially the same as the system (4), the only diﬀerence being that the second equation has been
multiplied through by 2. Any solution of the system (4) is a solution of the system (7), and vice versa.
In the general situation, it is not diﬃcult to see the following.
PROPOSITION 1A. (ELEMENTARY ROW OPERATIONS) Consider the array (2) corresponding
to the system (1).
(a) Interchanging the ith and j th rows of (2) corresponds to interchanging the ith and j th equations
in (1).
(b) Adding a multiple of the ith row of (2) to the j th row corresponds to adding the same multiple of
the ith equation in (1) to the j th equation.
(c) Multiplying the ith row of (2) by a nonzero constant corresponds to multiplying the ith equation
in (1) by the same nonzero constant.
In all three cases, the collection of solutions to the system (1) remains unchanged.
Let us investigate how we may use elementary row operations to help us solve a system of linear
equations. As a ﬁrst step, let us continue again with Example 1.1.4.
Example 1.2.4. Consider again the system of linear equations (8) x1 + 3x2 + x3 + 5x4 + x5 = 5,
x2 + x3 + 2x4 + x5 = 4,
2x1 + 4x2 + 7x4 + x5 = 3, represented by the array 1
0
2 (9) 3
1
4 1
1
0 5
2
7 1
1
1 5
4.
3 Let us now perform elementary row operations on the array (9). At this point, do not worry if you do
not understand why we are taking the following steps. Adding −2 times the ﬁrst row of (9) to the third
row, we obtain 13
1
5
1
5
0 1
1
2
1
4 .
0 −2 −2 −3 −1 −7
From here, we add 2 times the second row to the third row to obtain 1
0
0 (10) 3
1
0 1
1
0 5
2
1 1
1
1 5
4.
1 Next, we add −3 times the second row to the ﬁrst row to obtain 1
0
0 0 −2
11
00 −1
2
1 −2
1
1 −7
4 .
1 Next, we add the third row to the ﬁrst row to obtain 1
0
0
Chapter 1 : Linear Equations 0
1
0 −2
1
0 0
2
1 −1
1
1 −6
4 .
1
page 4 of 21 c Linear Algebra Finally, we add −2 times the third to row to 10
0 1
(11)
00 W W L Chen, 1982, 2005 the second row to obtain −2 0 −1 −6
1 0 −1
2 .
011
1 We remark here that the array (10) is said to be in row echelon form, while the array (11) is said to be
in reduced row echelon form – precise deﬁnitions will follow in Sections 1.5–1.6. Let us see how we may
solve the system (8) by using the arrays (10) or (11). First consider (10). Note that this represents the
system
x1 + 3x2 + x3 + 5x4 + x5 = 5,
x2 + x3 + 2x4 + x5 = 4, (12) x4 + x5 = 1.
First of all, take the third equation
x4 + x5 = 1.
If we let x5 = t, then x4 = 1 − t. Substituting these into the second equation, we obtain (you must do
the calculation here)
x2 + x3 = 2 + t.
If we let x3 = s, then x2 = 2 + t − s. Substituting all these into the ﬁrst equation, we obtain (you must
do the calculation here)
x1 = −6 + t + 2s.
Hence
x = (x1 , x2 , x3 , x4 , x5 ) = (−6 + t + 2s, 2 + t − s, s, 1 − t, t)
is a solution of the system (12) for every s, t ∈ R. In view of Proposition 1A, these are also precisely the
solutions of the system (8). Alternatively, consider (11) instead. Note that this represents the system
x1
(13) − 2x3
x2 + x3 − x5 = −6,
− x5 = 2,
x4 + x5 = 1. First of all, take the third equation
x4 + x5 = 1.
If we let x5 = t, then x4 = 1 − t. Substituting these into the second equation, we obtain (you must do
the calculation here)
x2 + x3 = 2 + t.
If we let x3 = s, then x2 = 2 + t − s. Substituting all these into the ﬁrst equation, we obtain (you must
do the calculation here)
x1 = −6 + t + 2s.
Hence
x = (x1 , x2 , x3 , x4 , x5 ) = (−6 + t + 2s, 2 + t − s, s, 1 − t, t)
Chapter 1 : Linear Equations page 5 of 21 c Linear Algebra W W L Chen, 1982, 2005 is a solution of the system (13) for every s, t ∈ R. In view of Proposition 1A, these are also precisely
the solutions of the system (8). However, if you have done the calculations as suggested, you will notice
that the calculation is easier for the system (13) than for the system (12). This is clearly a case of the
array (11) in reduced row echelon form having more 0’s than the array (10) in row echelon form, so that
the system (13) has fewer nonzero coeﬃcients than the system (12). 1.3. Row Echelon Form
Definition. A rectangular array of numbers is said to be in row echelon form if the following conditions
are satisﬁed:
(1) The leftmost nonzero entry of any nonzero row has value 1. These are called the pivot entries.
(2) All zero rows are grouped together at the bottom of the array.
(3) The pivot entry of a nonzero row occurring lower in the array is to the right of the pivot entry of
a nonzero row occurring higher in the array.
Next, we investigate how we may reduce a given array to row echelon form. We shall illustrate the
ideas by working on an example.
Example 1.3.1. Consider the array 0
0 0
0 0
2
1
1 5
4
2
2 0
7
3
4 Step 1: Locate the leftmost nonzero column and
illustration here, × denotes an entry that has been ×05
× 2 4 ×12
×12 15
1
0
1 5
3
.
1
2 cover all columns to the left of this column (in our
covered). We now have 0 15 5
7 1 3
.
301
412 Step 2: Consider the part of the array that remains uncovered. By interchanging rows if necessary,
ensure that the topleft entry is nonzero. So let us interchange rows 1 and 4 to obtain ×12412
× 2 4 7 1 3 .
×12301
× 0 5 0 15 5
Step 3: If the top entry on the leftmost uncovered column is a, then we multiply the top uncovered row
by 1/a to ensure that this entry becomes 1. So let us divide row 1 by 1 to obtain ×12412
× 2 4 7 1 3 !
×12301
× 0 5 0 15 5
Step 4: We now try to make all entries below the top entry on the leftmost uncovered column zero.
This can be achieved by adding suitable multiples of row 1 to the other rows. So let us add −2 times
row 1 to row 2 to obtain ×
× ×
×
Chapter 1 : Linear Equations 1
0
1
0 2
0
2
5 4
−1
3
0 1
−1
0
15 2
−1 .
1
5
page 6 of 21 c Linear Algebra Then let us add −1 times row 1 to row 3 ×
× ×
× to obtain Step 5: Now cover the top row. We then ×
× ×
× W W L Chen, 1982, 2005 obtain 1
0
0
0 ×
0
0
0 2
0
0
5 4
−1
−1
0 ××
0 −1
0 −1
5
0 1
−1
−1
15 2
−1 .
−1
5 ××
−1 −1 .
−1 −1
15
5 Step 6: Repeat Steps 1–5 on the uncovered array, and as many times as necessary so that eventually the
whole array gets covered. So let us continue. Following Step 1, we locate the leftmost nonzero column
and cover all columns to the left of this column. We now have ×××× × × × × 0 −1 −1 −1 .
× × 0 −1 −1 −1
××5
0
15
5
Following Step 2, we interchanging rows if necessary to ensure that the topleft entry is nonzero. So let
us interchange rows 1 and 3 (here we do not count any covered rows) to obtain ×××× × ×
0
15
5
× × 5 .
× × 0 −1 −1 −1
× × 0 −1 −1 −1
Following Step 3, we multiply the top row by a
leftmost uncovered column becomes 1. So let us ×××
× × 1 ××0
××0 suitable number to ensure that the top entry on the
multiply row 1 by 1/5 to obtain ×××
0
3
1
.
−1 −1 −1
−1 −1 −1 Following Step 4, we do nothing! Following Step 5, we ××××
× × × × × × 0 −1
× × 0 −1 cover the top row. We then obtain ××
× ×
.
−1 −1
−1 −1 Following Step 1, we locate the leftmost nonzero column and cover all columns to the left of this
column. We now have ×
× ×
× ×
×
×
× ×
×
×
× ×
×
−1
−1 ×
×
−1
−1 ×
×
.
−1
−1 Following Step 2, we do nothing! Following Step 3, we multiply the top row by a suitable number to
ensure that the top entry on the leftmost uncovered column becomes 1. So let us multiply row 1 by −1
to obtain ×
× ×
×
Chapter 1 : Linear Equations ×
×
×
× ×
×
×
× ×
×
1
−1 ×
×
1
−1 ×
×
.
1
−1
page 7 of 21 c Linear Algebra W W L Chen, 1982, 2005 Following Step 4, we now try to make all entries below the top entry on the leftmost uncovered column
zero. So let us add row 1 to row 2 to obtain ××××××
× × × × × × .
×××111
×××000
Following Step 5, we cover the top row. We then obtain ×
× ×
× ×
×
×
× ×
×
×
× ×
×
×
0 ×
×
×
0 ×
×
.
×
0 Following Step 1, we locate the leftmost nonzero column and cover all columns to the left of this
column. We now have ×
× ×
× ×
×
×
× ×
×
×
× ×
×
×
× ×
×
×
× 1
0
0
0 2
1
0
0 4
0
1
0 1
3
1
0 ×
×
.
×
× Step ∞. Uncover everything! We then have 0
0 0
0 2
1
,
1
0 in row echelon form.
In practice, we do not actually cover any entries of the array, so let us repeat here the same argument
without covering anything – the reader is advised to compare this with the earlier discussion. We start
with the array 0
2
1
1 5
4
2
2 0
7
3
4 15
1
0
1 5
3
.
1
2 1
2
1
0 0
0 0
0 2
4
2
5 4
7
3
0 1
1
0
15 2
3
.
1
5 Interchanging rows 1 and 4, we obtain 0
0 0
0 Adding −2 times row 1 to row 2, and adding −1 times row 1 to row 3, we obtain 0
0 0
0 1
0
0
0 2
0
0
5 4
−1
−1
0 1
−1
−1
15 2
−1 .
−1
5 1
0
0
0 2
5
0
0 4
0
−1
−1 1
15
−1
−1 2
5
.
−1
−1 Interchanging rows 2 and 4, we obtain 0
0 0
0
Chapter 1 : Linear Equations page 8 of 21 c Linear Algebra W W L Chen, 1982, 2005 Multiplying row 1 by 1/5, we obtain 0
0 0
0 1
0
0
0 2
1
0
0 4
1
0
3
−1 −1
−1 −1 2
1
.
−1
−1 1
0
0
0 2
1
0
0 4
0
1
−1 2
1
.
1
−1 Multiplying row 3 by −1, we obtain 0
0 0
0 1
3
1
−1 Adding row 3 to row 4, we obtain 0
0 0
0 1
0
0
0 2
1
0
0 4
0
1
0 1
3
1
0 2
1
,
1
0 in row echelon form.
Remarks. (1) As already observed earlier, we do not actually physically cover rows or columns. In any
practical situation, we simply copy these entries without changes.
(2) The steps indicated the the ﬁrst part of the last example are for guidance only. In practice, we
do not have to follow the steps above religiously, and what we do is to a great extent dictated by good
common sense. For instance, suppose that we are faced with the array
2
3 3
2 2
0 1
2 . If we follow the steps religiously, then we shall multiply row 1 by 1/2. However, note that this will
introduce fractions to some entries of the array, and any subsequent calculation will become rather
messy. Instead, let us multiply row 1 by 3 to obtain
6
3 9
2 6
0 3
2 . 6
6 9
4 6
0 3
4 . Then let us multiply row 2 by 2 to obtain Adding −1 times row 1 to row 2, we obtain
6
0 9
−5 63
−6 1 . In this way, we have avoided the introduction of fractions until later in the process. In general, if we start
with an array with integer entries, then it is possible to delay the introduction of fractions by omitting
Step 3 until the very end.
Chapter 1 : Linear Equations page 9 of 21 c Linear Algebra Example 1.3.2. Consider the array 2
1
3 1
3
2 3
2
0 2
4
0 W W L Chen, 1982, 2005 5
1.
2 Try following the steps indicated in the ﬁrst part of the previous example religiously and try to see how
complicated the calculations get. On the other hand, we can modify the steps with some common sense.
First of all, we interchange rows 1 and 2 to obtain 13241
2 1 3 2 5.
32002
The reason for taking this step is to put an entry 1 at the top left without introducing fractions anywhere.
When we next add multiples of row 1 to the other rows to make 0’s below this 1, we do not introduce
fractions either. Now adding −2 times row 1 to row 2, we obtain 1
0
3 3
−5
2 2
−1
0 4
−6
0 1
3.
2 Adding −3 times row 1 to row 3, we obtain 13 0 −5
0 −7 2
−1
−6 4
−6
−12 1
3 .
−1 2
7
−6 4
42
−12 1
−21 .
−1 2
7
30 4
42
60 Next, multiplying row 2 by −7, we obtain 1
0
0 3
35
−7 Multiplying row 3 by −5, we obtain 1
0
0 3
35
35 1
−21 .
5 Note that here we are essentially covering up row 1. Also, we have multiplied rows 2 and 3 by suitable
multiples to that their leading nonzero entries are the same, in preparation for taking the next step
without introducing fractions. Now adding −1 times row 2 to row 3, we obtain 1
0
0 3
35
0 2
7
23 4
42
18 1
−21 .
26 Here, the array is almost in row echelon form, except that the leading nonzero entries in rows 2 and 3
are not equal to 1. However, we can always multiply row 2 by 1/35 and row 3 by 1/23 if we want to
obtain the row echelon form 13 2
4
1 0 1 1/5
6/5
−3/5 .
00 1
18/23 26/23
If this diﬀers from the answer you got when you followed the steps indicated in the previous example
religiously, do not worry. row echelon forms are not unique!
Chapter 1 : Linear Equations page 10 of 21 c Linear Algebra W W L Chen, 1982, 2005 1.4. Reduced Row Echelon Form
Definition. A rectangular array of numbers is said to be in reduced row echelon form if the following
conditions are satisﬁed:
(1) The leftmost nonzero entry of any nonzero row has value 1. These are called the pivot entries.
(2) All zero rows are grouped together at the bottom of the array.
(3) The pivot entry of a nonzero row occurring lower in the array is to the right of the pivot entry of
a nonzero row occurring higher in the array.
(4) Each column containing a pivot entry has 0’s everywhere else in the column.
We now investigate how we may reduce a given array to reduced row echelon form. Here, we basically
take an extra step to convert an array from row echelon form to reduced row echelon form. We shall
illustrate the ideas by continuing on an earlier example.
Example 1.4.1. Consider again the array 0
0 0
0 0
2
1
1 5
4
2
2 0
7
3
4 15
1
0
1 5
3
.
1
2 We have already shown in Example 1.3.1 that this array can be reduced to row echelon form 0
0 0
0 1
0
0
0 2
1
0
0 4
0
1
0 1
3
1
0 2
1
.
1
0 Step 1: Cover all zero rows at the bottom of the array. We now have 0
0 0
× 1
0
0
× 2
1
0
× 4
0
1
× 1
3
1
× 2
1
.
1
× Step 2: We now try to make all the entries above the pivot entry on the bottom row zero (here again we
do not count any covered rows). This can be achieved by adding suitable multiples of the bottom row
to the other rows. So let us add −4 times row 3 to row 1 to obtain 0 1 2 0 −3 −2
3
1
0 0 1 0 .
0001
1
1
××××× ×
Step 3: Now cover the bottom row. We then obtain 0
0 ×
× 1
0
×
× 2
1
×
× 0 −3
0
3
××
×× −2
1
.
×
× Step 4: Repeat Steps 2–3 on the uncovered array, and as many times as necessary so that eventually
the whole array gets covered. So let us continue. Following Step 2, we add −2 times row 2 to row 1 to
obtain 0
0 ×
×
Chapter 1 : Linear Equations 1
0
×
× 0
1
×
× 0 −9
0
3
××
×× −4
1
.
×
×
page 11 of 21 c Linear Algebra W W L Chen, 1982, 2005 Following Step 3, we cover row 2 to obtain 0
× ×
× 1
×
×
× 0
×
×
× 0 −9
××
××
×× −4
×
.
×
× Following Step 2, we do nothing! Following Step 3, we cover row 1 to obtain ×
× ×
× ×
×
×
× ×
×
×
× ×
×
×
× ×
×
×
× ×
×
.
×
× Step ∞. Uncover everything! We then have 0
0 0
0 1
0
0
0 0
1
0
0 0
0
1
0 −9
3
1
0 −4
1
,
1
0 in reduced row echelon form.
Again, in practice, we do not actually cover any entries of the array, so let us repeat here the same
argument without covering anything – the reader is advised to compare this with the earlier discussion.
We start with the row echelon form 012412
0 0 1 0 3 1 .
000111
000000
Adding −4 times row 3 to row 1, we obtain 0
0
1
0 −3
3
1
0 −2
1
.
1
0 0100
0 0 1 0 0001
0000 −9
3
1
0 −4
1
,
1
0 0
0 0
0 1
0
0
0 2
1
0
0 Adding −2 times row 2 to row 1, we obtain in reduced row echelon form. 1.5. Solving a System of Linear Equations
Let us ﬁrst summarize what we have done so far. We study a system (1) of m linear equations in n
variables x1 , . . . , xn . If we omit reference to the variables, then the system (1) can be represented by
the array (2), with m rows and n + 1 columns. We next reduce the array (2) to row echelon form or
reduced row echelon form by elementary row operations.
By Proposition 1A, the system of linear equations represented by the array in row echelon form or
reduced row echelon form has the same solution set as the system (1). It follows that to solve the system
Chapter 1 : Linear Equations page 12 of 21 c Linear Algebra W W L Chen, 1982, 2005 (1), it remains to solve the system represented by the array in row echelon form or reduced row echelon
form. We now describe a simple way to obtain all solutions of this system.
Definition. Any column of an array (2) in row echelon form or reduced row echelon form containing a
pivot entry is called a pivot column.
First of all, let us eliminate the situation when the system has no solutions. Suppose that the array
(2) has been reduced to row echelon form, and that this contains a row of the form
0 ... 0 1 n corresponding to the last column of the array being a pivot column. This row represents the equation
0x1 + . . . + 0xn = 1;
clearly the system cannot have any solution.
Definition. Suppose that the array (2) in row echelon form or reduced row echelon form satisﬁes the
condition that its last column is not a pivot column. Then any variable xi corresponding to a pivot
column is called a pivot variable. All other variables are called free variables.
Example 1.5.1. Consider the array 0
0 0
0 1
0
0
0 0
1
0
0 −4
1
,
1
0 −9
3
1
0 0
0
1
0 representing the system
− 9x5 = −4, x2
x3 + 3x5 =
x4 + x5 = 1,
1. Note that the zero row in the array represents an equation which is trivial! Here the last column of the
array is not a pivot column. Now columns 2, 3, 4 are the pivot columns, so that x2 , x3 , x4 are the pivot
variables and x1 , x5 are the free variables.
To solve the system, we allow the free variables to take any values we choose, and then solve for the
pivot variables in terms of the values of these free variables.
Example 1.5.2. Consider the system of 4 linear equations (14) 5x3
+ 15x5 = 5,
2x2 + 4x3 + 7x4 + x5 = 3,
x2 + 2x3 + 3x4
= 1,
x2 + 2x3 + 4x4 + x5 = 2, in the 5 variables x1 , x2 , x3 , x4 , x5 . If we omit reference to the variables, then the system can be represented by the array (15) Chapter 1 : Linear Equations 0
0 0
0 0
2
1
1 5
4
2
2 0
7
3
4 15
1
0
1 5
3
.
1
2
page 13 of 21 c Linear Algebra W W L Chen, 1982, 2005 As in Example 1.3.1, we can reduce the array (15) to row echelon form (16) 0
0 0
0 1
0
0
0 2
1
0
0 4
0
1
0 1
3
1
0 2
1
,
1
0 representing the system (17) x2 + 2x3 + 4x4 + x5 = 2,
x3
+ 3x5 = 1,
x4 + x5 = 1. Alternatively, as in Example 1.4.1, we can reduce the array (15) to reduced row echelon form (18) 0
0 0
0 1
0
0
0 0
1
0
0 0
0
1
0 −9
3
1
0 −4
1
,
1
0 representing the system
− 9x5 = −4, x2
(19) x3 + 3x5 =
x4 + x5 = 1,
1. By Proposition 1A, the three systems (14), (17) and (19) have exactly the same solution set. Now, we
observe from (16) or (18) that columns 2, 3, 4 are the pivot columns, so that x2 , x3 , x4 are the pivot
variables and x1 , x5 are the free variables. If we assign values x1 = s and x5 = t, then we have, from
(17) (harder) or (19) (easier), that
(20) (x1 , x2 , x3 , x4 , x5 ) = (s, 9t − 4, −3t + 1, −t + 1, t). It follows that (20) is a solution of the system (14) for every s, t ∈ R.
Example 1.5.3. Let us return to Example 1.2.4, and consider again the system (8) of 3 linear equations in
the 5 variables x1 , x2 , x3 , x4 , x5 . If we omit reference to the variables, then the system can be represented
by the array (9). We can reduce the array (9) to row echelon form (10), representing the system (12).
Alternatively, we can reduce the array (9) to reduced row echelon form (11), representing the system
(13). By Proposition 1A, the three systems (8), (12) and (13) have exactly the same solution set. Now,
we observe from (10) or (11) that columns 1, 2, 4 are the pivot columns, so that x1 , x2 , x4 are the pivot
variables and x3 , x5 are the free variables. If we assign values x3 = s and x5 = t, then we have, from
(12) (harder) or (13) (easier), that
(21) (x1 , x2 , x3 , x4 , x5 ) = (−6 + t + 2s, 2 + t − s, s, 1 − t, t). It follows that (21) is a solution of the system (8) for every s, t ∈ R.
Example 1.5.4. In this example, we do not bother even to reduce the matrix to row echelon form.
Consider the system of 3 linear equations (22) Chapter 1 : Linear Equations 2x1 + x2 + 3x3 + 2x4 = 5,
x1 + 3x2 + 2x3 + 4x4 = 1,
3x1 + 2x2
= 2,
page 14 of 21 c Linear Algebra W W L Chen, 1982, 2005 in the 4 variables x1 , x2 , x3 , x4 . If we omit reference to the variables, then the system can be represented
by the array 21325
1 3 2 4 1.
(23)
32002
As in Example 1.3.2, we can reduce the array (23) to the form 13
2
4
1 0 35 7 42 −21 ,
(24)
0 0 23 18
26
representing the system
x1 + 3x2 + 2x3 + 4x4 =
1,
35x2 + 7x3 + 42x4 = −21,
23x3 + 18x4 = 26. (25) Note that the array (24) is almost in row echelon form, except that the pivot entries are not 1. By
Proposition 1A, the two systems (22) and (25) have exactly the same solution set. Now, we observe
from (24) that columns 1, 2, 3 are the pivot columns, so that x1 , x2 , x3 are the pivot variables and x4 is
the free variable. If we assign values x4 = s, then we have, from (25), that
(26) (x1 , x2 , x3 , x4 ) = 16
28 24
19 18
26
s + ,− s − ,− s + ,s .
23
23 23
23 23
23 It follows that (26) is a solution of the system (22) for every s ∈ R. 1.6. Homogeneous Systems
Consider a homogeneous system of m linear equations of the form (27) a11 x1 + a12 x2 + . . . + a1n xn = 0,
a21 x1 + a22 x2 + . . . + a2n xn = 0,
.
.
.
am1 x1 + am2 x2 + . . . + amn xn = 0, with n variables x1 , x2 , . . . , xn . If we omit reference to the variables, then system (27) can be represented
by the array a12
a22
.
.
. ...
... a1n
a2n
.
.
. 0
0
.
.
. am1 (28) a11 a21
.
.
. am2 ... amn 0 of all the coeﬃcients.
Note that the system (27) always has a solution, namely the trivial solution
x1 = x2 = . . . = xn = 0.
Indeed, if we reduce the array (28) to row echelon form or reduced row echelon form, then it is not
diﬃcult to see that the last column is a zero column and so cannot be a pivot column.
Chapter 1 : Linear Equations page 15 of 21 Linear Algebra c W W L Chen, 1982, 2005 On the other hand, if the system (27) has a nontrivial solution, then we can multiply this solution
by any nonzero real number diﬀerent from 1 to obtain another nontrivial solution. We have therefore
proved the following simple result.
PROPOSITION 1B. The homogeneous system (27) either has the trivial solution as its only solution
or has inﬁnitely many solutions.
The purpose of this section is to discuss the following stronger result.
PROPOSITION 1C. Suppose that the system (27) has more variables than equations; in other words,
suppose that n > m. Then there are inﬁnitely many solutions.
To see this, let us consider the array (28) representing the system (27). Note that (28) has m rows,
corresponding to the number of equations. Also (28) has n + 1 columns, where n is the number of
variables. However, the column of (28) on the extreme right is a zero column, corresponding to the
fact that the system is homogeneous. Furthermore, this column remains a zero column if we perform
elementary row operations on the array (28). If we now reduce (28) to row echelon form by elementary
row operations, then there are at most m pivot columns, since there are only m equations in (27) and m
rows in (28). It follows that if we exclude the zero column on the extreme right, then the remaining n
columns cannot all be pivot columns. Hence at least one of the variables is a free variable. By assigning
this free variable arbitrary real values, we end up with inﬁnitely many solutions for the system (27). 1.7. Application to Electrical Networks
A simple electric circuit consists of two basic components, electrical sources where the electrical potential
E is measured in volts (V ), and resistors where the resistence R is measured in ohms (Ω). We are
interested in determining the current I measured in amperes (A).
The electrical potential between two points is sometimes called the voltage drop between these two
points. Currents and voltage drops can be positive or negative.
The current ﬂow in an electrical circuit is governed by three basic rules:
• Ohm’s law: The voltage drop E across a resistor with resistence R with a current I passing through
it is given by E = IR.
• Current law: The sum of the currents ﬂowing into any point is the same as the sum of the currents
ﬂowing out of the point.
• Voltage law: The sum of the voltage drops around any closed loop is equal to zero.
Around any loop, we select a positive direction – clockwise or anticlockwise as we see ﬁt. We have the
following convention:
• The voltage drop across a resistor is taken to be positive if the current ﬂows in the positive direction
of the loop, and negative if the current ﬂows in the negative direction of the loop.
• The voltage drop across an electrical source is taken to be positive if the positive direction of the
loop is from + to −, and negative if the positive direction of the loop is from − to +.
We shall explain this again in our examples.
Chapter 1 : Linear Equations page 16 of 21 c Linear Algebra W W L Chen, 1982, 2005 Example 1.7.1. Consider the electric circuit shown in the diagram below:
8Ω A
/ /
I3 I1
+
20V I2 − 4Ω 20Ω +
B − 16V We wish to determine the currents I1 , I2 and I3 . Applying the Current law to the point A, we obtain
I1 = I2 + I3 . Applying the Current law to the point B , we obtain the same. Hence we have the linear
equation
I1 − I2 − I3 = 0.
Next, let us consider the left hand loop, and let us take the positive direction to be clockwise. By
Ohm’s law, the voltage drop across the 8Ω resistor is 8I1 , while the voltage drop across the 4Ω resistor is
4I2 . On the other hand, the voltage drop across the 20V electrical source is negative, since the positive
direction of the loop is from − to +. The Voltage law applied to this loop now gives 8I1 + 4I2 − 20 = 0,
and we have the linear equation
8I1 + 4I2 = 20, or 2I1 + I2 = 5. Next, let us consider the right hand loop, and let us take the positive direction to be clockwise. By Ohm’s
law, the voltage drop across the 20Ω resistor is 20I3 , while the voltage drop across the 4Ω resistor is
−4I2 . On the other hand, the voltage drop across the 16V electrical source is negative, since the positive
direction of the loop is from − to +. The Voltage law applied to this loop now gives 20I3 − 4I2 − 16 = 0,
and we have the linear equation
4I2 − 20I3 = −16, or I2 − 5I3 = −4. We now have a system of three linear equations
I1 − I2 − I3 =
(29) 0, 2I1 + I2 = 5,
I2 − 5I3 = −4. The augmented matrix is given by 1 −1 −1
2 1
0
0 1 −5 0
5 ,
−4 with reduced row echelon form 1
0
0 0
1
0 0
0
1 2
1.
1 Hence I1 = 2 and I2 = I3 = 1. Note here that we have not considered the outer loop. Suppose again that
we take the positive direction to be clockwise. By Ohm’s law, the voltage drop across the 8Ω resistor is
8I1 , while the voltage drop across the 20Ω resistor is 20I3 . On the other hand, the voltage drop across
the 20V and 16V electrical sources are both negative. The Voltage law applied to this loop then gives
8I1 + 20I3 − 36 = 0. But this equation can be obtained by combining the last two equations in (29).
Chapter 1 : Linear Equations page 17 of 21 c Linear Algebra W W L Chen, 1982, 2005 Example 1.7.2. Consider the electric circuit shown in the diagram below:
8Ω A
/ / I1 I3
O +
20V I2 − 6Ω 8Ω +
B 5Ω − 30V We wish to determine the currents I1 , I2 and I3 . Applying the Current law to the point A, we obtain
I1 + I2 = I3 . Applying the Current law to the point B , we obtain the same. Hence we have the linear
equation
I1 + I2 − I3 = 0.
Next, let us consider the left hand loop, and let us take the positive direction to be clockwise. By Ohm’s
law, the voltage drop across the 8Ω resistor is 8I1 , the voltage drop across the 6Ω resistor is −6I2 , while
the voltage drop across the 5Ω resistor is 5I1 . On the other hand, the voltage drop across the 20V
electrical source is negative, since the positive direction of the loop is from − to +. The Voltage law
applied to this loop now gives 8I1 − 6I2 + 5I1 − 20 = 0, and we have the linear equation
13I1 − 6I2 = 20.
Next, let us consider the outer loop, and let us take the positive direction to be clockwise. By Ohm’s
law, the voltage drop across the 8Ω resistor on the top is 8I1 , the voltage drop across the 8Ω resistor
on the right is 8I3 , while the voltage drop across the 5Ω resistor is 5I1 . On the other hand, the voltage
drop across the 30V and 20V electrical sources are both negative, since the positive direction of the loop
is from − to + in each case. The Voltage law applied to this loop now gives 8I1 + 8I3 + 5I1 − 50 = 0,
and we have the linear equation
13I1 + 8I3 = 50.
We now have a system of three linear equations (30) The augmented matrix is given by 1
1 −1
0 13 −6 0
20 ,
13 0
8
50 I1 + I2 − I3 = 0,
13I1 − 6I2
= 20,
13I1
+ 8I3 = 50. with reduced row echelon form 1
0
0 0
1
0 0
0
1 2
1.
3 Hence I1 = 2, I2 = 1 and I3 = 3. Note here that we have not considered the right hand loop. Suppose
again that we take the positive direction to be clockwise. By Ohm’s law, the voltage drop across the
8Ω resistor is 8I3 , while the voltage drop across the 6Ω resistor is 6I2 . On the other hand, the voltage
drop across the 30V electrical source is negative. The Voltage law applied to this loop then gives
8I3 + 6I2 − 30 = 0. But this equation can be obtained by combining the last two equations in (30).
Chapter 1 : Linear Equations page 18 of 21 c Linear Algebra W W L Chen, 1982, 2005 Problems for Chapter 1
1. Consider the system of linear equations
2x1 + 5x2 + 8x3 = 2,
x1 + 2x2 + 3x3 = 4,
3x1 + 4x2 + 4x3 = 1.
a) Write down the augmented matrix for this system.
b) Reduce the augmented matrix by elementary row operations to row echelon form.
c) Use your answer in part (b) to solve the system of linear equations.
2. Consider the system of linear equations
4x1 + 5x2 + 8x3 = 0,
x1
+ 3x3 = 6,
3x1 + 4x2 + 6x3 = 9.
a) Write down the augmented matrix for this system.
b) Reduce the augmented matrix by elementary row operations to row echelon form.
c) Use your answer in part (b) to solve the system of linear equations.
3. Consider the system of linear equations
x1 − x2 − 7x3 + 7x4 = 5,
−x1 + x2 + 8x3 − 5x4 = −7,
3x1 − 2x2 − 17x3 + 13x4 = 14,
2x1 − x2 − 11x3 + 8x4 = 7.
a) Write down the augmented matrix for this system.
b) Reduce the augmented matrix by elementary row operations to row echelon form.
c) Use your answer in part (b) to solve the system of linear equations.
4. Solve the system of linear equations
x + 3y − 2z = 4,
2x + 7y + 2z = 10. 5. For each of the augmented matrices below, reduce the matrix to row echelon or reduced row echelon
form, and solve the system of linear equations represented by the matrix: 11 2 1
5
12
3 −3 1
6 3 2 −1 3
a) b) 2 −5 −3 12 2 4 3 1 4 11
7
71
8
5
2 1 −3 2
1
6. Reduce 1
0
a) 0
0 1
c) 2
3 each of the following arrays by elementary row operations 2345
110
2 3 4 5
0 1 1
b) 0345
001
0045
000 11 21 31 41 51
12 22 32 42 52 13 23 33 43 53 Chapter 1 : Linear Equations to
0
0
1
1 reduced row echelon form: 0
0 0
1 page 19 of 21 c Linear Algebra W W L Chen, 1982, 2005 7. Consider a system of linear equations in ﬁve variables x = (x1 , x2 , x3 , x4 , x5 ) and expressed in matrix
form Ax = b, where x is written as a column matrix. Suppose that the augmented matrix (Ab)
can be reduced by elementary row operations to the row echelon form 1
0 0
0 3
0
0
0 2
1
0
0 0
1
1
0 4
1
.
7
0 6
2
1
0 a) Which are the pivot variables and which are the free variables?
b) Determine all the solutions of the system of linear equations.
8. Consider a system of linear equations in ﬁve variables x = (x1 , x2 , x3 , x4 , x5 ) and expressed in matrix
form Ax = b, where x is written as a column matrix. Suppose that the augmented matrix (Ab)
can be reduced by elementary row operations to the row echelon form 1
0 0
0 2
1
0
0 0
3
0
0 3
1
1
0 5
3
.
4
0 1
2
1
0 a) Which are the pivot variables and which are the free variables?
b) Determine all the solutions of the system of linear equations.
9. Consider the system of linear equations
x1 + λx2 − x3 =
1,
2x1 + x2 + 2x3 = 5λ + 1,
x1 − x2 + 3x3 = 4λ + 2,
x1 − 2λx2 + 7x3 = 10λ − 1.
a) Reduce its associated augmented matrix to row echelon form.
b) Find a value of λ for which the system is soluble.
c) Solve the system.
10. Consider the electric circuit shown in the diagram below:
A
o
o
I1 I3 10Ω I2 − 5Ω − + 20V 6Ω B + 40V Determine the currents I1 , I2 and I3 . You must explain each step carefully, quoting all the relevant
laws on electric circuits. 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 carries the correct sign.
Chapter 1 : Linear Equations page 20 of 21 c Linear Algebra W W L Chen, 1982, 2005 11. Consider the electric circuit shown in the diagram below:
A
o
o
I1 I3
O 8Ω I2 − 1Ω 8Ω − +
B 60V + 20V Determine the currents I1 , I2 and I3 . You must explain each step carefully, quoting all the relevant
laws on electric circuits. 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 carries the correct sign.
12. Consider the electric circuit shown in the diagram below:
8Ω A 20Ω B / / I1 I3
− +
50V − I2 5Ω o I5 C 5Ω I4
o + 10V I6 D Determine the currents I1 , I2 , I3 , I4 , I5 and I6 . You must explain each step carefully, quoting all the
relevant laws on electric circuits. 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 carries the correct sign. Chapter 1 : Linear Equations page 21 of 21 ...
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This note was uploaded on 06/13/2009 for the course TAM 455 taught by Professor Petrina during the Fall '08 term at Cornell University (Engineering School).
 Fall '08
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