Linear Algebra full course lecture notes

LECTURE ONE
lecturer one


'


introduction to linear algebra
-



complex numbers



Quadratic equations
'
general formula →
ax + be + C =
0

( real
coefficients a. b. C ,
a # o )



by
'



at
x + sc + = 0




'



( %) ( Ea)
'

✗ + -
+
£ =o




( Ea)
'
e-
E)
+ 2

⇐
= -




a




'
B2-4AC
µ tea )
+ =




Za




B2-4AC
tea
x + =




Za




0C = -

b In B2-4AC

2A




assuming discriminant
B2-4AC is non
negative
-




complex numbers


'

=L = -1 introduces an
imaginary unit ,
and set


it = -
I i = A


complex numbers are written in the Cartesian form


and real
✗ +
iy where
2
sc
y are numbers and
i = -

I



oc =
Re 1-2)

1m It )
y
=




both are real numbers .




all real numbers a c- IR are uicluded in the set € of all

complex numbers saying a = a + io

, and real
iy
the number 2- × + where × are
y
=
,




can be represented as a point on the co -
ordinate plane .




atm
subtracting
2- = x +
iy
y
- -




→
- -




is also like the subtraction of
;
planar vectors .




→
so the

Z,
Zz
-




=/ x
,
+
iy ) ,
-

(x >
+
iy ) ,


Addition
ly yz )
= (x ,
-

✗
a) + i
,
-




is like addition of planar
^
1M
vectors .




I oc
, y ) + ( x2 , y
) =
( x
,
+ ✗
z , y tyz )
, , , ,




Z2
1SC ,
+
iy ) ,
+ ( sci +
iyz )
=
(x ,
+ x2 ) t i
ly ,
+
yz )
^


2-
-22
-




,




geometrically we use i
£

parallelogram
,
the rule s

nlm Re

-9
Z
, +2-2
multiplication
-




try
-


2- , = 0C , , - l

'

s
starts with the bracket

>
"
expansion
2-2=34 tiyz
s


Re Z
,
Z
,
=
( ×
,
tiy ) ( ,
x
,
+
iyz )


= I
,x , t ×
, iy ,
t ✗
ziy ,
t
ity , Yz

( i ? 1) -




regroup .




the expressions in the brackets are =
4C , sci
-




y , y , ) + i /× ,
y ,
+
Kay ) ,


real numbers hence the result is a
,

→
complex number




main properties of the three operations

commlltativity -2 t
2-2 Zz 2- -2
2-2 2-2-2
=
-




t =
, I , ,




It
"


2)
"


associativity =L
(2-2+2-3) + 2- + Z
-


+ =

, , }




distribute vity -
Z
,
I Zz t 2- 3) = Z
, Zz t Z ,
Z}

, LECTURE TWO
lecture three



complex conjugations nim



2- = ✗ +
iy
let 2- = x +
iy . its complex conjugate y
- - - -
-
-
-•
,
defined I
is as
iy
=
x -




t

od
'
Re
the operation 2- → É is complex conjugation .
I


its !
geometrically ,
the reflection of the complex -




y
- - - - -
- - -




in real axis
I = x
-




iy
plane the




properties
a- É =
-2 →
the double conjugation is the identity transformation

* if É =
2- then Z is real .




* the product É -2 = (x -




iy ) ( xtiy ) = ✗
'
+
y
'
is
always real and

non
negative
-




.




Division
mm



±
how to define
22
while keeping all standard properties of the

division ?


In particular ,
we want to be able to write the result in the form
a + ib
,
a
,
b t IR which would confirm that the result is a complex
number .




2-
,
Z
,
✗ Éz -2
,
= ×
,
+
iy , Zz
=
SC2 +
iyz
-
=




-22 2-2 ✗ Éz


= 2-
,
✗ Éz

xi +
y;



?
'
É
2-2=1×2 iya ) / xztiyz ) ; liyz ) i' ? ?
y;
-

=
x =
oc =
x +
y
- -




,




is real and positive if 2-2 =/ 0




modulus and
argument

the
writing 2- =
set
iy is the Cartesian form of the complex number 2- .




The modulus ( or absolute value ) of a complex number



2- = × +
iy is 12-1 =


,
✗
2
+
y
'

,* we denote the modulus also by r= 12-1 ( radius )
* this is the distance from the origin to the point 2- =

octiy
* remember É 2- =
✗
2
+
yz = 12-12
*
Always remember that 12-1 is real and positive ( unless 2- =o )
a 1M



2- =
octiy
y →
-
- - - -




# I
=\
g 1


! >
a Re




the
argument arglz ) of a complex number 2- = ✗ +
iy is the

angle between the positive real semi -
axis and the direction
from the origin to the point Z .




the
argument 0 is defined up to an integer number of
rotations
full about the origin .




Any integer multiple of 21T
may
be added to 0 to produce another admissible value of the

/ argument multi valued )
argument . The is

The
argument is measured in radians .




A 1M




y
- - -
- - -




→
2- = ✗ +
iy notice the
arglz ) when 2- =o is not

I defined .




'
't
=\ ,
✗
I in
general if oc =/ 0 then tano =¥
,



a) =
angle) I
,


×
The


However ,
a- tan
"
(1) is not
always true .




indeed the range of the function tan
"
is the interval ( I
-




,
E)
1¥ )
'
F- tan valid and 4th quadrants
-




therefore is for the 1st
only ,




that is , for so > 0 .




for the 2nd and 3rd quadrants ( the left half -




plane that is ✗ < 0 )
1¥ )
"
the simplest is to take D= tan + IT .

, LECTURE THREE
lecture three



The
projections of the position rector of the complex number E-
xtiy are :




✗ = r cos it
y=rsino

so we can write 2- = rcosotirsuno-lztlcosotisi.no)


this is called the
trigonometric form or polar form of a complex number .




^
'M



→
2-
=x+iy

I
r
y=rsino
,
'
✗ Re
=rcoso




The trigonometric form of a complex number is expressed in terms of its modulus

and
argument .




2- = 12-1 ( cos at isino )



The Cartesian form 2- =
✗ +
iy is better for addition or subtraction whereas


trigonometric ( or exponential) form is better for multiplication or division . .




further properties

* -2+7 = É + Ñ

* In = E I

* Z1J =
E1J
* I 2- I =
IE1
*
/ zf =
ZÉ
a- / ZW1 = I2-1IW /

a- I2-1WI =/ 2- I / IW1


complex numbers on the unit circle

The unit circle on the complex plane is the circle of radius 1 with its

centre at the origin .




Therefore numbers on it are those of modulus 1 : t = 12-1=1
so 2- = cosotis.no



Let us
multiply two numbers such Z and W with arguments 0 and ✗ .




ZW =/ Cos it + isino )( cos ✗ + isinx )

Costco > icososinxticosxsindtiisinos.int arglzw )
=

× + Thus

=
( Costco > x
-

Sino Sino ) tilcososinxtcosxs.int ) =
arglz) +
arg ( w
)
=
cos lots ) + i sin to + x )