1
I.
VECTOR SPACE
Consider a two dimensional plane, every point in the place can be mapped to a unique vector. All of the vectors
form a linear vector space
v
with a dimensionality equal to two.
Any vector can be labeled as (
x, y
) which can be written as
(
x, y
) =
x
(1
,
0) +
y
(0
,
1)
(1)
The sum of two vectors satisfies,
a
(
x
1
, y
1
) +
b
(
x
2
, y
2
) = (
ax
1
+
bx
2
+
ay
1
+
by
2
)
(2)
In this vector space, we can only define a dot product,
(
x
1
, y
x
)
·
(
x
2
, y
2
) = (
x
1
x
2
, y
2
y
2
)
(3)
Notice that (1
,
0)
·
(0
,
1) = 0 and (1
,
0)
·
(1
,
0)=1, which we call the two vectors are orthogonal to each other. Since
any vector can be written as a linear combination of the two orthogonal vectors, we call the linear vector space
v
has
dimension two and (1
,
0) and (0
,
1) form a complete orthonormal basis of
V
.
II.
DIRAC NOTATION
The similar idea can be generalized to a space with any dimensions. We call a general linear vector space
V
with
dimension N if any vector in
V
can be written as a linear combination of N orthonormal vectors. We will use a ket
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 Fall '11
 NA
 Linear Algebra, Vector Space, Hilbert space, oij

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