10
1 Some Linear Algebra
For example, if we consider the complex vector space C
2 , then the canonical basis
for C
2 is given by the vectors
|v 1 ≡
1
0
and |v 2 ≡
0
1
.
Another basis of C
2 is the basis consisting of the vectors
|w 1 ≡ 1/
√
2
1
1
and |w 2 ≡ 1/
√
2
1
−1
.
If
|v ≡
⎡
⎢
⎢
⎢
⎣
z 1
z 2
. . .
z n
⎤
⎥
⎥
⎥
⎦
,
then its dual vector v| (known as bra in the Dirac notation) is the row vector v| =
[z
∗
1 z
∗
2 · · · z
∗
n ].
In the following, we define inner product on vector spaces over C (as mentioned
above, this is the scenario of quantum mechanics).
Definition 1.7.1 Let V be a vector space over C. An inner product on V is a function
(·, ·) : V × V −→ C that satisfies the following conditions:
(i) ∀ |v, |w i ∈ V , and ∀ c i ∈ C, i = 1, 2, . . . , n, one has
|v,
n
i=1
c i |w i
=
n
i=1
c i (|v, |w i );
(ii) ∀ |v, |w ∈ V , (|v, |w) = (|w, |v)
∗ , where ∗ denotes the complex conjugate of the complex number (|w, |v);
(iii) ∀ |v ∈ V , (|v, |v) ≥ 0 and (|v, |v) = 0 ⇐⇒ |v = 0.
A vector space V endowed with an inner product is said to be an inner product space.
Definition 1.7.2 An inner product space V is said to be a Hilbert space if it is a
complete (with respect to the metric defined by the inner product) inner product space.
The metric defined by the inner product is given by d(v, w) =
√
v − w, v − w for
all v, w ∈ V , where v − w, v − w = (|v − w, |v − w)
Remark 1.7.1 (1) If |v, |w are vectors in V then they are called orthogonal if
(|v, |w) = 0. An orthogonal basis for V is a basis B V = {|v i |i ∈ S} of V such that
|v i is orthogonal to |v j for every i = j, i, j ∈ S. If, in addition, the vectors |v j are
unit vectors (see the line after Definition 1.7.3 for the definition of unit vector) then
B V is said to be an orthonormal basis.