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Vector spaces over C

9
...
2

is given in the case where the scalars are

real numbers
...
Thus, the vector
space axioms make sense if we allow the scalars to be the set of complex numbers
...


EXERCISE 1

2 {[�� ] Z1,Z2 c},

Le t C

=

I

E

with addition defined by

+ wi]
[21]Z2 + [w']
[21
W2 Z2+W2
=

and scalar multiplication by

a

E

C defined by

a2
[zi]
[
a z2 az21]
=

Show that C2 is a vector space over C
...
We begin by considering the set of complex numbers C itself as a vector
space
...
That is, every complex number can be written in the form a1, where a is a
complex number
...
Alternatively, we could choose to use the basis {i}
...

Another way of looking at this is to observe that when we use complex scalars, any
two non-zero elements of the space IC are linearly dependent
...

z2
It follows that with respect to complex scalars, a basis for IC must have fewer than two
dimensions
...
Hence, viewed as a vector space over JR, the set of complex numbers is
two-dimensional, with "standard" basis {l , i}
...
2, we sometimes write complex numbers in a way that
exhibits the property that IC is a two-dimensional real vector space: we write a complex
number z in the form
z = x + iy = (x,y) = x(l,0) + y(O, 1)
Note that (1, 0) denotes the complex number 1 and that (0, 1) denotes the complex
number i
...
, which justifies our work with the complex plane in Section 9
...
However, notice
that this representation of the complex numbers as a real vector space does not include
multiplication by complex scalars
...


Definition

The vector space IC11 is defined to be the set

C"

with addition of vectors and scalar multiplication defined as above
...

Since the complex conjugate is so useful in C, we extend the definition of a com­
plex conjugate to vectors in C'1•

z1

Definition

Complex Conjugate

The

complex conjugate of z=

[]
:

E

en is defined to be l =

Zn

EXAMPLE 1
Let z=

1 +i

1 +i

1-i

-2i

-2i

2i


...
We say that
the complex numbers if for any

L is linear over

a EC and v1, v2 EV we have

L(av1 + v2) = aL(v1) + L(v2)
We can also define subspaces just as we did for real vector spaces, and the range
and nullspace of a linear mapping will be subspaces of the appropriate vector spaces,
as before
...
]
1 -

_

L is
[L] =

The image of z=

-2i
i ,

]
[
0) =
l

[

1 +i
2

-2i
1- i

1+2i

]

3+i

under Lis calculated by

l

[
[
J
=
=
L
(x)
z
L

1+i
2

-2i
...
i
3+t

+2i

i

1-


...
)

[L], and

0
...


EXAMPLE3

[

1

Let A = 11
...


1 +i

-i

-1 + 2i

l

l

;

- i
...
That is, a basis for Row(A) is

,

-1

-[

m l [j: Ul

We next recall that the columns of A corresponding to the columns of
tain leading 1' fonn a basis for Col(A)
...


Using the

reduced row echelon form of A, we find that the homogeneous system is equivalent to
Zt

+ iz2

- Z4

Z3 - iz4

The free variables are
z1

= -is+t and

z3

z2

and

z4,

so we let

Z2
Z3
Z4

Thus, a basis for Null(L) is

Let A

1 +i
-[

nullspace of A
...
Then we get

= it
...
Find a basis for the rowspace, columnspace, and

Complex Multiplication as a Matrix Mapping
We have seen that Ccan be regarded as a real two-dimensional vector space
...
We first
consider a special case
...


[o -1]
1

0

That is, multiplication by i corresponds to a rotation by an

y

...
3
...


More generally, we can consider multiplication of complex numbers by a complex
number

a

= x +yi
...

b
a
a

[a ]

Ma

of JR2 with

PROBLEMS 9
...


Al Calculate the following
...


1

A3 Find a basis for the rowspace, columnspace, and
nullspace of the following matrices
...


-1

l

1

1+ 2i

2

-2

-2 + 3i
-2 +i

1+i

-1

-l+i

C2 such that

--

L(l,0)=

[� ]
2i

1

and

L(O, 1) =

I

-1 -i

[ � � �]

Homework Problems

[ ] [- ]

Bl Calculate the following
...


2 - �i
1- l

-

-l

3 + 2;

(b)

B3 Determine which of the following sets is a basis
for C3•

2 +

-�

- 3 +

�

1+ 3t

1-

l

-2

(a)

(b)

(c) -3i

(d) (-1 _i)

2

B4 Find a basis for the rowspace, columnspace, and

3i

nullspace of the following matrices
...


(a) A=

1

1

i

i

i

1

l+i

l+i

-1-i

1 i

2 -i

...

(d) D

�

[�

-1

2

i

]

[I� Tl
i

6

8

4;

o

-4i

Conceptual Problems
Dl (a) Prove that multiplication by any complex num­
ber a = a + bi can be represented as a lin­
ear mapping Ma of IR2 with standard matrix

[� �]
-


...


(c) Verify the result by calculating Mo: for a =

3

-

4i and interpreting it as in part (b)
...


C(2, 2) is a complex vector space
...


D3 Define isomorphisms for complex vector spaces
and check that the arguments and results of Sec­
tion

4
...


D4 Let {v\,v2,v3} be a basis for JR3
...
V2, v3}

is also a basis for

plex vector space)

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