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Lecture notes Mathematics II (MATH2011A) - ALGEBRA_Chapter_4 $21.01
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Lecture notes Mathematics II (MATH2011A) - ALGEBRA_Chapter_4
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  • Course
  • Mathematics II (MATH2011A)
  • Institution
  • University Of The Witwatersrand (wits)

This document clearly describes, with detailed notes and examples, how to evaluate/solve the following: ~ Dot products and Orthonormal Bases ~ Unitary and Hermitian Matrices ~ Fourier series as taught by the University of the Witwatersrand. As a student, I am always searching for a great se...

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Document information

  • February 14, 2022
  • 68
  • 2021/2022
  • Class notes
  • Darlison nyirenda
  • All classes

Subjects

  • orthonormality
  • dot products
  • orthonormal bases
  • orthonormal
  • unitary
  • hermitian
  • matrix
  • matrices
  • fourier series
  • unitary matrices
  • hermitian matrices
  • length of a vector
  • matrix p
  • properties of the dot product

Written for

  • University of the Witwatersrand (wits)
  • Mathematics II (MATH2011A)
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Content preview

CHAPTER 4 : ORTHONOMALITY :


DOT PRODUCTS AND ORTHONORMAL BASES :




0
DEALT WITH VECTORS IN ☐2h 0 VECTOR WITH N COMPONENTS AND THEIR

COMPONENTS ARE REAL NUMBERS .




TO FIND THE DOT PRODUCT
IN 12h : : MULTIPLY RESPECTIVE COMPONENTS
GIVEN TWO VECTORS AND ADD THE PRODUCTS




0 HOW DO WE PERFORM THE DOT PRODUCT IF GIVEN VECTORS WITH COMPONENTS

IN THE SET OF COMPLEX NUMBERS ?

" ^
% EXTEND THE DOT PRODUCT FROM VECTORS IN DZ TO VECTORS ②
.




DEFINITION : THERE ARE TWO VECTORS A AND b
• "
¥wo•÷÷i¥: Let (a b ( b , ; bz ; bn)
a
; Az ; ; an ;
= =
, . . . .
; . . .




h COMPONENTS OF EACH VECTOR
A b E ② (
0


WHERE
j ARE COMPLEX NUMBERS



THEN THE DOT PRODUCT OF VECTORS A AND b IS DENOTED
BY THE USUAL NOTATION A • b AND DEFINED BY :




J¥¥Ñ"
IN SUMMATION




v
V V L
TO COMPUTE DOT
CONJUGATION
PRODUCT
BETW . 2 VECTORS THAT INTRODUCE
LIE ☒ CONJUGATE N
!
IN

②
• i.

THE COMPONENTS IN THE SECOND SINCE
VECTOR !




EXAMPLE : LET VECTOR a =
(1; i ;3 j b = ( i ;it1 ; 1
COMPUTE THE DOT PRODUCT .




i. aob =
( I ;i ; 3) ( i ;i+1;1 •

* WHEN CONJUGATING A complex

= (1) ( j ) + ( i)( it 1) + (3) (1) NUMBER
OF THE
JUST CHANGE
IMAGINARY PART !
THE SIGN



=
(1) C-i) till i ) + -
3
= i ti- IZ + 3 - i2= -1

= 4
,


NB : IF A ; b E ☐2h :O EACH COMPONENT OF VECTORS 9 and b ARE REAL
DEFINITION DOT PRODUCT
con "
"YÉ☐n NUMBERS OF
'


. .
. .
REDUCES TO THE USUAL ONE KNOW .




h

.
'


.
a •
b = aibi BUT since bi =
bi
i=1
n
= aibi
i=1
D

, IF
°
NOTE : TAKE DOT PRODUCT OF VECTOR A BY ITSELF :

n

ao a =
aiai
[ =L COMPLEX NUMBER
MULTIPLIED BY ITS
D- CONJUGATE


2
BUT KNOW FROM COMPLEX NUMBERS : 2 =
Z
n
2
÷ a o a =
, ai
i=1
I.
COMPONENT OF
A VECTOR !

% CAN NOW DEFINE THE
LENGTHOF THE VECTOR
!

^
DEFINITION : THE LENGTH OF A VECTOR A WHERE A E ② IS
GIVEN BY
• + VE SQUARE ROOT
OF DOT PRODUCT




=
a aoaa


EXAMPLE : VECTOR a = (i; it 1 ; -1
FIND THE LENGTH OF VECTOR A .




1 A / =
A- •
a ( BY DEFINITION



( i ;i+1 ; 1) ( i ;i+1 ; -1) SHORTCUT :
2 a. a
•
= -




n 2
( i)( T) + (it 1) ( its )t C- 1)C- a. a =
=
ai
i=1
= it ( it 1) ( 1- i) + 1
-




n =3 :
=
It i i 2+1 i +1 -
-




=
4
=
il 't it , 2+112
, ANY COMPLEX
NUMBER CAN BE
WRITTEN IN THIS FORM
:
ÉÉEAÉNEN
2
SQUARED !
✗
tiy A MODULUS :
(Re)2t(Im)Z
LENGTH OF
g. -12 xzty !
: a 1 (2) Zt I
=
=
z a = 4 = +
p
=
4
,




NB PROPERTY SUPPOSE A MEANS THAT IT 'S
'
: : = 0 . .
LENGTH IS ZERO .




IF GIVEN VECTOR SUCH THAT ITS LENGTH IS ZERO THEN SURELY THE
,
VECTOR IS ZERO :

LENGTH :


To A = 0 IF AND ONLY IF THE VECTOR ITSELF A = 0 .




( WORKS BOTH WAYS !)

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