Algebra Chapter 3

CHAPTER 3


ALGEBRA
LINEAR ALGEBRA

, '

Linear Spaces uneven

Gm
linear of
'




A collection
things
'




space is a



usually vectors .




⇒
_mvA
Property the things
e. satisfy

{ IN
Notation
s IER }
:

=
=
'

uch
where the that

things came
from
"


Read :
S is a set of vectors in 3- dimensions

equals
"


such that mad V are




*
A linear space satisfies two main properties

1 If YES then eyes
i
we s is ceased under
say
.




scalar meet i plication .




2 If if and ya Es then Y +
Yz E S ;
say
we
.




,



S is closed under addition
( different of
a
just way
defining a linear operator

,*
Examples of linear spaces
:




"

1 .

Be the space of all real numbers
in the nth dimension .




¢
n

the space of all complex numbers
in the nth dimension



2 .
Real valued functions of x.
Complex valued functions of a .




3
7
.
The space of possible inputs at a


linear operator ( Domain )


The space of outputs of a


linear operator ( Range )


5 The null space of a linear operator

6 Zero
by itself is a trivial linear operator

The )
"

collection of vectors in IR 1=14 . . .vn

such that v. =o ; s =
{ I =w ,
...

a) 1 v. =o
} 2
id est S { I ( o vz .vn ) I VERN
}
= .
= .


,




NIT { k ( V. vn ) IV. =L
} ?
=


8 s = . . .




.




9 Functions with period 2T

id est f ( x ) = f ( set 2T )
Cf y
Because is also 2T .


periodic .




and ftg is also 2T -

periodic

, Example
Is the of all CZ.ws such that Ztiw=o linear ?
space , ,




ie is s
{ (z ) lztiwto } linear ?
=

.
,
w




Let lz ,
w ) Es ,
Is CCZ ,
w ) ES ?
→ check C I Z ,
w ) =
( Cz ,
C w )



RHS(
= CZTICCW )
=
C Z + iw )
=
C Cos
o
good
=




Let (Z , ,
W , ) and ( Zz ,
Wa ) ES and consider

(Z } ,
W
} ) =
( Zi ,
W .
) t ( Zz ,
Wz )

→ check Zs + iw }
=
( Z ,
+ iwi ) + ( Zz +
iwa )

Rtts = (Z it iwi ) t ( zz town )
=
0 + 0

a
good
=




S linear
'


. .
is