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Math 20F Linear Algebra
In Linear Algebra we solve systems of linear equations but linear algebra also
provides a framework in which to think about many problems.
Lecture 1: 1.1 Linear systems of equations.
A
linear system
of
m
equations
in
n
unknowns
is of the form:
(1.1)
a
11
x
1
+
a
12
x
2
+
...
+
a
1
n
x
n
=
b
1
a
21
x
1
+
a
22
x
2
+
...
+
a
2
n
x
n
=
b
2
.
.
.
a
m
1
x
1
+
a
m
2
x
2
+
...
+
a
mn
x
n
=
b
m
where the
a
ij
’s and
b
i
’s are given constants and
x
1
,x
2
,...,x
n
are unknowns to be
determined. It is called and
m
×
n
system
. A
solution
to the system is an ordered
n
tuple (
x
1
,x
2
,...,x
n
) such that all the
m
equations are satisﬁed. The set of all
solutions are called the
solution set
.
Let us try to understand the geometric meaning of a general 2
×
2
systems
:
a
11
x
1
+
a
12
x
2
=
b
1
a
21
x
1
+
a
22
x
2
=
b
2
The solutions to each of the equations form a line in the (
x
1
,x
2
)plane.
(
x
1
,x
2
) is therefore a solution to the system if and only if it lies on both these lines.
Ex 1
Find all solutions to the system
n
x
1
+
x
2
= 3
2
x
1

x
2
= 0
Sol
The two lines in the plane intersect at the point (1
,
2), which is the only solution
Ex 2
Find all solutions to the system
n
x
1
+
x
2
= 3
2
x
1
+ 2
x
2
= 0
Sol
The lines are parallel so they don’t intersect. No solutions!
Ex 3
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