Algebra Chapter 4

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