Lecture notes Mathematics II (MATH2011A) - ALGEBRA_Chapter_4
11 views 0 purchase
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...
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 !)
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller Akshay101. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $21.54. You're not tied to anything after your purchase.
