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Part I. Linear Algebra
1.1 Linear Equations; Some Geometry
A
linear (algebraic) equation in
n
unknowns
,
x
1
,x
2
,.
.
.
n
, is an equation of the form
a
1
x
1
+
a
2
x
2
+
···
+
a
n
x
n
=
b
where
a
1
,a
2
.
.
n
and
b
are given numbers called the
coeﬃcients
. In particular
ax
=
b
is a linear equation in one unknown;
ax
+
by
=
c
is a linear equation in two unknowns (if
a
and
b
are real numbers, not both 0, then the graph
of the equation is a straight line); and
ax
+
by
+
cz
=
d
is a linear equation in three unknowns (if
a, b
and
c
are real numbers, not all 0, then the graph
is a plane in 3space).
Our main interest in this treatment of linear algebra is solving systems of linear equations.
Remark:
In a general study of
Linear Algebra
the coeﬃcients and the values of the unknowns
are assumed to come from some given Feld
F
. In the treatment here we will use the Feld of real
numbers,
R
; the term “number” means “real number.”
±
Linear equations in one unknown.
We begin with simplest case: one equation in one unknown.
If you were asked to Fnd a real number
x
such that
ax
=
b
you would probably say “that’s easy,”
x
=
b
a
. But the fact is, this “solution” is not necessarily
correct. ±or example, consider the three equations
(1)
2
x
=6
,
(2) 0
x
,
(3)
0
x
=0
.
±or equation (1), the solution
x
/
2 = 3 is correct. However, consider equation (2); there is no
real number that satisFes this equation! Now look at equation (3);
every
real number satisFes (3).
In general, it is easy to see that for the equation
ax
=
b
, exactly one of three things happens:
(a) There is precisely one solution (
x
=
b/a
, when
a
±
= 0).
(b) There are no solutions (
a
,b
±
= 0).
1
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View Full Document(c) There are infnitely many solutions (
a
=
b
= 0).
As you will see, this simple case illustrates (and motivates) the general situation.
For any system of
m
linear equations in
n
unknowns, exactly one of three possibilities occurs: a unique solution, no
solution, or in±nitely many solutions
.
Linear equations in two unknowns
We begin with the one equation:
ax
+
by
=
c.
Here we are looking For ordered pairs oF real numbers (
x,y
) which satisFy the equation. IF
a
=
b
=0
and
c
±
= 0, then there are no solutions. IF
a
=
b
=
c
= 0, then
every
ordered pair (
) satisfes
the equation; there are infnitely many solutions, the whole
xy
plane, a twodimensional set. IF at
least one oF
a
and
b
is diﬀerent From 0, then the equation
ax
+
by
=
c
represents a straight line
in the
xy
plane and the equation has infnitely many solutions, the set oF all points on the line, this
time a onedimensional set. Note that For one linear equation in two unknowns it is not possible to
have a unique solution; we either have no solution or infnitely many solutions.
Two linear equations in two unknowns is a more interesting case. IF
a
and
b
are not both zero,
and
c
and
d
are not both zero, then the pair oF equations
ax
+
by
=
α
cx
+
dy
=
β
represents a pair oF lines in the
xy
plane. We are looking For ordered pairs (
) oF numbers that
satisFy both equations simultaneously. As you know, two lines in the plane either
(a) have a unique point oF intersection (this occurs when the lines have diﬀerent slopes), or
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 Spring '08
 Staff
 Linear Algebra, Geometry, Linear Equations, Equations

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