100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
Previously searched by you
Previously searched by you
logo
Algebra Chapter 4 R50,00
Add to cart
Quickly navigate to
Preview
  2.  
  3. University Of The Witwatersrand (wits)
  4.  
  5. Mathematics II

Study guide

Algebra Chapter 4
 0 purchase

This chapter includes notes on Orthonormality, Unitary and Hermitian Matrices, Quadratic Forms and Fourier Series. Each section has multiple worked examples. Notes were taken on an iPad.

Last document update: 5 year ago

Preview 4 out of 40  pages

Document information

  • August 18, 2019
  • August 20, 2019
  • 40
  • 2018/2019
  • Study guide

Subjects

Written for

All documents for this subject (22)

Seller

avatar-seller
romario25

Reviews received

Content preview

CHAPTER 4

ALGEBRA
ORTHO NORMALITY

, Dot products and orthonormal bases
In this section want to investigate the geometrical aspect of vectors
°

we


especially lengths and angles ,
in higher dimensions
°
We
generate.se the dot product to n dimensions and define :




A-
ok =
( a ,
Az
,
.
. .


an ) o ( bi .bz ,
. . .
bn )
BTt BI bi where b- the conjugate of b
=
a + a
. . . t an is
,




if bi
=
ai

,




( -3 ) find
example :
If =
( Iti ,
2-
Ii 3 )
be , and =
ti
,
H2i ,
-
i :




II , b- a-Cda '
,
and b-ok


fo I = ( Hi
,
2 -

is 3) of -3 ti
,
It Li
,
-
i )
=
( it ill -3 -
i ) t
( 2. i ) ( I -
zi ) t (3) Li )
= -




3
-


i -

3 i + I t 2 -

4 i -

i -
2 -13 i
= -



2 - 6 i



b-eat =
I -
3 ti ,
It 2 is -
i ) o
( Hi ,
2 -

i , 3)
=
( -3 ti ) (I -

i ) t It -12 i ) .
12 ti ) t fi ) (3)
*
the of a ok
2+6 i
conjugate
= -




b-ok 16
=




16
Eobe =

,For complex numbers the factor essential
conjugate in the second are


because the dot product of a vector with itself must be the


square of its magnitude
'




.




Is
"



a- a . = Ex Akai =

19kt which is real and non -




negative
, ,




i. e
Ibet =
EE


orthogonal ( ! )
n

If ok 0 Freiman
a then vectors and be
=

a are




If a
and b- are raw vectors then the dot product is equal to a


f bi I
-




matrix product i.e a. be
=


.
a
[ transpose of b conjugate



example :
I
=
I Hi
,
2 ti
,
3 ) and be =
f- 3 ti
,
It 2i ,
-
i )
a-of
= -

2 -


Gi from above



so a b-
a =

[ Hi ,
zti .
3 ] of :{if ? mtaitriglg
multiplication



=
2 Gi
-
-




.
-




It follows from matrix algebra that

I i) Ia +
d) . I =
Eohe + do b-

( ii ) I Ce ) a
be =
C Ceobe )
} 778¥
Iii it be a
=
a y
ate ,
b.




.

For lil and Iii ) show that the dot product is a linear operator
different
For column vectors A and B the dat product is given slightly
.

a

matrix product : A o
B =
ATE

, Root iiis




Required to
prove
that :
q.ie =

Ee


LHS =
la ,
,
as
.
.
.




an ) o
(b ,
,
ba -
- -

bn )


=

AT + asbT +
. . .
+ Anton
=
a- , b t a-zbz t - a .
t AT bn

be a- RHS
=
=
.




F④ ,




Jefinition : An orthonormal basis of linear is basis
p
a space a

g.
consisting of mutually perpendicular unit vectors


example : if t =
GO ,
o
) , j =
I 0,1 ,
o ) and I =
6,0 ,
I ) show that

bi ) forms
ii. j .
an orthonormal basis .




I oj =
( I , 0,0 ) s
( a ,
I
, a) =
0+0+0=0


I °
bi =

( I ,
o ,
a ) o
( a ,
a ,
I ) - a -10+0=0


Jabi =
( oil ,
o ) .
( o ,
o
, 1)
.



-
O -10+0=0




lil =
III =
Ik I =
EE = I




'


I i. j bi ) orthonormal
. .

,
is an basis
Ba

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

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

Quick and easy check-out

You can quickly pay through EFT, credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

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 this summary from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller romario25. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy this summary for R50,00. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

70057 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy summaries for 15 years now

Start selling

Recently viewed by you

Essay ·

(2)

Portfolio stage 3

Class notes ·

(0)

Full summary MBS

Summary ·

(0)

For Oom Piet IEB

Summary ·

(0)

Summary virology

Summary ·

(0)

Kippenpracticum 4

Summary ·

(1)

All Cases BBS1001
R50,00
  • (0)
Add to cart
Added