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Title: Inner products in complex vector spaces
Description: Linear algebra course
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Inner Products in
Complex Vector
Spaces

9
...

Our first thought would be to determine if we can extend the dot product to e11•
Does this define an inner product on e11? Let z x iJ, then we have

+

=

Z Z = Z1 + +
= (� +
...
- y�) +2i(X1Y1 +
...
0
does not even make sense
...
fining
an inner product in en
...
We recall that if z E e, then zZ lzl2 ?:'
...
Hence, it
makes sense to choose
...
/(z,

=

w>= z· w

as this gives us

(z,Z'J
Definition

(

In en the standard inner product , ) is defined by

Standard Inner Product on C"

(z,

EXAMPLE 1

= z· t= Z1Z1 +· · · +ZnZn = [zi[2 +· · · +[ Zn [2 ?:'
...
+ZnWn,

[ �]
...


Solution:

= v
...
(2 - i)il = r-� : � ] (2 [;: �]
= r-�:�Jc2+i)[���]
= (2+i)(3+
= +23i
i

2i)(2

i)(l - i)
4

-1

·

-1


...


- i)

Observe that this does not satisfy the properties of the real inner product
...


EXERCISE 1

Let it=

[i � ]
2i

and v=

[i � ��l

Determine (it, v), (2iil, v), and (it, 2iv)
...


Definition

Let V be a vector space over C
...


Note that property (1) allows us to define the (standard) length by llz ll = (z, z)112,
as desired
...
Notice that if all the
vectors are real, the Hermitian property simplifies to symmetry
...
However,
this property reduces to bilinearity when the scalars are all real
...
How­
ever, new proofs are required
...
Then, for all w,

z EV,
(4) l(z, w)I ::
...
:; llzll + llwll

Proof:

We prove (4) and leave the proof of (5) as Problem Dl
...

Let C(m, n) denote the complex vector space of m x n matrices with complex en­
tries
...
11111• Of course, we want to define

the standard inner product on C(m, n) in a similar way
Title: Inner products in complex vector spaces
Description: Linear algebra course
Buy These NotesPreview