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
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
