MAT1341A: LECTURE NOTES FOR FALL 2008
MONICA NEVINS (HEAVILY BORROWING FROM HANDWRITTEN NOTES OF BARRY JESSUP)
These are my lecture notes for MAT1341A, Fall 2008. I am following a combination of my
own notes from Fall 2007, and Barry Jessup’s handwritten notes, and the book by Nicholson
[N]
1.
First class: Complex Numbers
1.1.
Introduction.
•
Present syllabus, introduce webpage, virtual campus and textbook.
•
Describe large class interactions: ask questions, respect your peers
•
Course goals:
–
Understanding where and how linear algebra applies (vector spaces, linear maps,
n
dimensions; engineering, computer science, physics, economics),
–
also visavis Calculus (linear approximations work amazingly well in the real
world);
–
generalizations to
n
dimensions
1.2.
Complex Numbers.
[N, 2.5.1, 2.5.2, 2.5.4, 2.5.6]
In brief, one might say that algebra is the study of solutions of polynomial equations. Linear
algebra is then the study of solutions to linear equations. (It gets interesting when you allow
multiple variables and multiple equations, but we’ll get to that later.)
For today, let’s look at an application of Algebra, as a “welcome back to algebra” for every
one after a long summer away, and as a heads up to our Engineers (particularly Electrical
Engineers) and Physicists who will soon be using complex numbers all the time.
It begins with the equation
x
2
+ 1 = 0
.
We know this has no solutions, since every square is positive. (However, note that in ancient
times one would have said that
x
+1 = 0 has no solutions, either, since negative numbers don’t
represent quantities.) So let us denote one solution of this equation by
i
, for “imaginary”
(notation thanks to Euler, 1777):
i
2
=

1
or
i
=
√

1
.
Then apply the normal rules of algebra, such as
√

9 =
9
·
(

1) =
√
9
·
√

1 = 3
i.
Date
: September 4, 2008.
1
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MONICA NEVINS (HEAVILY BORROWING FROM HANDWRITTEN NOTES OF BARRY JESSUP)
(Please note that here, as in Calculus, we adhere to the convention that
√
a
2
=

a

, that is,
the answer is the one positive square root, and not a choice of them.)
Now
i
alone is not quite enough. Consider the equation
x
2
+ 4
x
+ 8 = 0
.
By the quadratic formula, its roots are the two numbers
x
=

4
±
√
16

32
2
=

2
±
1
2
√

16 =

2
±
1
2
(4
i
) =

2
±
2
i.
Let’s check that this makes sense:
Plug
x
= (

2 + 2
i
) into the quadratic equation and
simplify:
(

2 + 2
i
)
2
+ 4(

2 + 2
i
) + 8 = (4

4
i

4
i
+ 4
i
2
) + (

8 + 8
i
) + 8 = 4 + 4
i
2
= 0
where in the last step we remembered that
i
2
=

1. Similarly we can check that

2

2
i
is
also a root.
With these thoughts (and the quadratic formula) in mind, we make the following definition.
Definition.
The set of
complex numbers
is the set
C
=
{
a
+
bi
:
a, b
∈
R
}
.
(Read this as: “the set of all things of the form
a
+
bi
where
a
and
b
are real numbers”.)
When we write
z
=
a
+
bi
∈
C
(read as:
z
(or
a
+
bi
) belongs to
C
), then
a
is called the
real part
of
z
(denoted
Re
(
z
)
) and
b
is called the imaginary part of
z
(denoted
Im
(
z
)
).
Note that
Re
(
z
)
and
Im
(
z
)
are real
numbers!
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 Winter '10
 DAIGLE
 Complex Numbers, Complex number, Barry Jessup

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