Lecture notes

Linear Algebra full course lecture notes

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Module
Linear Algebra (MATHEMATICS)

Institution
The University Of Liverpool (UoL)

Summary and notes on the full course of Linear Algebra at university level. In depth annotations and descriptions covering the whole semester. Consisting of all 28 lectures. Includes: determinants matrices gauss elimination reduction formula cramers rule etc.

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Uploaded on January 16, 2022
Number of pages 85
Written in 2021/2022
Type Lecture notes
Professor(s) Victor goryunov
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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 )

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