Quantum Mechanics for Mathematicians

Definition(Dual vector space). ForVa vector space over a fieldk, the dual
vector spaceV∗is the vector space of all linear mapsV→k, i.e.,

V∗={l:V→ksuch thatl(αv+βw) =αl(v) +βl(w)}

forα,β∈k, v,w∈V.

Given a linear transformationLacting onV, we can define:

Definition(Transpose transformation).The transpose ofLis the linear trans-
formation
Lt:V∗→V∗

given by
(Ltl)(v) =l(Lv) (4.1)

forl∈V∗,v∈V.

For any choice of basis{ej}ofV, there is a dual basis{e∗j}ofV∗that
satisfies
e∗j(ek) =δjk

Coordinates onVwith respect to a basis are linear functions, and thus elements
ofV∗. The coordinate functionvjcan be identified with the dual basis vector
e∗jsince
e∗j(v) =e∗j(v 1 e 1 +v 2 e 2 +···+vnen) =vj

It can easily be shown that the elements of the matrix forLin the basisejare
given by
Ljk=e∗j(Lek)

and that the matrix for the transpose map (with respect to the dual basis) is
the matrix transpose
(LT)jk=Lkj
Matrix notation can be used to write elements

l=l 1 e∗ 1 +l 2 e∗ 2 +···+lne∗n∈V∗

ofV∗as row vectors (
l 1 l 2 ··· ln

)

of coordinates onV∗. Evaluation oflon a vectorvis then given by matrix
multiplication

l(v) =

(

l 1 l 2 ··· ln

)











v 1

v 2

..

.

vn











=l 1 v 1 +l 2 v 2 +···+lnvn

For any representation (π,V) of a groupGonV, we can define a corre-
sponding representation onV∗: