Chapter 2- LINEAR EQUATIONS
LECTURE 4
Prof. Dr. Zafer ASLAN
LINEAR EQUATIONS
Introduction
The theory of linear equations plays an important and motivating role in the
subject of linear algebra. IN fact, many problems in linear algebra are equivalent
to studying a system of linear equations, e.g. Finding the kernel of liner mapping
and characterizing the subspace spanned by a set of vectors.
For simplicity, we assume that all equations in this chapter are over the real filed
R. We emphasize that the results and techniques also hold for equations over
the complex field C or over any arbitrary field K.
LINEAR EQUATIONS
Linear Equation
By a linear equation over the real field R, we mean an expression of the form:
a1x1 + a2x2 + ...+ anxn = b
(1)
where;
ai, b R and xi are in determinants (or unknowns or variables).
The scalars ai are called the coefficients of the xi respectively, and b is called the
constant term or simply constant of the equation. A set of values for the
unknowns, say ;
x1 = k1, x2=k2,...., xn=kn
is a solution of (1) if the statement obtained by substituting ki for xi. This set of
values is then said to satisfy the equation, then we denote the solution: u = (k1, k2,
..., kn)
LINEAR EQUATIONS
Linear Equation
Solutions of the equation (1) can be easily described and obtained.
There are three cases:
Case (i): One of the coefficients in (1) is not zero, say a10, then we can
rewrite the equations as follows:
a1x1=b-a2x2-...-anxn
or
x1=a1-1b-a1-1a2x2-a1-1anxn
Linear equation in one unknown:
ax = b, with a0 has the unique solution x = a-1b
LINEAR EQUATIONS
Linear Equation
Case (ii): All the coefficient in (1) are zero, and the constant is also zero.
That is the equation is of the form:
0x1+0x2+...+0xn =b, with b0
Then the equation has no solution.
Case (iii): All the coefficients in (1) are zero, and the constant is also
zero. That is the equation is of the form:
0x1+0x2+...+0xn=0
Then every n – tuple of scalars in R is a solution of the equation.
LINEAR EQUATIONS
System of Linear Equations
We now consider a system of m linear equations in the n unknowns
x1,..., xn:
a11x1+a12x2+...+a1nxn =b1
a21x1+a22x2+...+a2nxn =b2
.............................................
am1x1+am2x2+...+amnxn =bn
Then the equation has no solution.
Where aij, bi belong to the real field R.
LINEAR EQUATIONS
System of Linear Equations
The system is said to be homogeneous if the constants b1,..., bmn are all 0.
An n- tuple u) (k1, ...., kn) of real numbers is a solution (or a particular
solution) if it satisfies each of the equations; the set of all such solutions is
termed the solution set or the general solution.
The system of linear equations:
a11x1+a12x2+...+a1nxn =0
a21x1+a22x2+...+a2nxn =0
.........................................
am1x1+am2x2+...+amnxn =0
is called the homogeneous system associated with. The above system
always has a solution, namely the zero n- tuple 0=(0,0,...,0) called the zero
or trivial solution. Any other solution, if it exists, is called a nonzero or
nontrivial solution.
LINEAR EQUATIONS
System of Linear Equations
Theorem 2.1: Suppose u is a particular solution of nonhomogeneous
system and suppose W is the general solution of associated homogeneous
system. Then;
u+W = {u+w: w W}
is the general solution of the nonhomogeneous system.
LINEAR EQUATIONS
Solution of a System of Linear Equations
Consider the above system of linear equations. We reduce it to a simpler
system as follows:
Step 1.
Interchange equations so that the first unknown xi has a
nonzero coefficient in he first equation, that is, so that ai10.
Step 2.
For each i1, apply the operation.
Li -ai1Li + a11Li
That is, replace the ith linear equation Li by the equation obtained by
multiplying the first equation Li by –ai1, multiplying the ith equation Li by ai1, an
then adding.
LINEAR EQUATIONS
Solution of a System of Linear Equations
We then obtain the following system:
a11x1+a’12x2+...+a’1nxn =b’1
+a’2j2xj2+...+a’2nxn =b’2
.....................................
+a’mj2x2+...+a’mnxn =b’m
where a110. Here xj2 denotes the first unknown with a nonzero coefficient in an
equation other than the first; by Step 2, xj2 x1. This which eliminates an
unknown from succeeding equations is known as (Gauss) elimination.
LINEAR EQUATIONS
Solution of a System of Linear Equations
Theorem 2.2: The solution of the system in echelon form is as follows. There
are two cases:
i) r = n. That is there are as many equations as unknowns. Then the system
has a unique solution.
ii) rn. That is, there are fewer equations than unknowns. Then we can
arbitrarily assign values to the n-r free variables and obtain a solution of the
system.
In view of Theorem 2.1, the unique solution above can only occur when
the associated homogeneous system has only the zero solution.
LINEAR EQUATIONS
Solution of a System of Linear Equations
System of linear
equations
Inconsistent
No
solution
Consistent
Unique
solution
More than one
solution
LINEAR EQUATIONS
Solution of Homogeneous System of Linear Equations
If we begin with a homogeneous system of linear equations, then the system is
clearly consistent since, for example, it has the zero solution 0=(0,0,...,0). Thus
it can always be reduced to an equivalent homogeneous system in echelon
form.
a11x1+a12x2+...+a1nxn =0
+a2j2xj2+...+a2nxn=0
............................
amj2x2+...+amnxn =0
Hence we have the two possibilities:
(i) r = n. Then the system has only the zero solution.
(ii) rn. Then the system has a nonzero solution.
Theorem 2.3: A homogeneous system of linear equations with
more unknowns than equations has a nonzero solution.
Reference
Seymour LIPSCHUTZ, (1987): Schaum’s Outline of Theory
and Problems of LINEAR ALGEBRA, SI (Metric) Edition,
ISBN: 0-07-099012-3, pp. 334, McGraw – Hill Book Co.,
Singapore.
Next Lecture (Week 5)
Chapter 3: MATRICES