2: Duality and normal transformations
The dual of a vector space V over a field F is the vector space Vd =
£(V,F) of linear functional f : V -+ F. If V is of finite dimension
and B = {v\,..., vn} is a basis of V then the basis that is dual to B is
Bd = {vf,..., v*} where each vf : V -» F is given by
For every x G V we have
a? = vf(x)vi +
and for every / £ Vd we have
If (v,) n , (^t)n are ordered bases of V and (r^) n , (wf )n the correspond-
ing dual bases then the transition matrix from (vf ) n to (wf ) n is (-P"1)*
where P is the transition matrix from (v,) n to (ty,) n . In particular,
consider V = \Rn. Note that if
is a basis of IRn then the transition matrix from B to the canonical basis
(c,*)n of \Rn is M = [wiy]ftXn where mt-y = ayt-. The transition matrix
from J5d to (e^)n is given by (M~1)t. We can therefore usefully denote
the dual basis by
, 2: Duality and normal transformations
where [an,..., a t n ] denotes the ith row of M~l, so that
• • - + ainxn.
The fadua/ of an element xisxA :Vd -> F where z A (t/ d ) = yd(x). It
is common practice to write yd{x) as (x, yd) and say that j / d annihilates
x if (z, t/d) = 0. For every subspace W of V the set
= { / G 7 d | (Vx G W) (z, / ) = 0}
is a subspace of W and
1
=dim7.
The transpose of a linear transformation f : V —> W is the linear
mapping /* : Wd -* V d described by yd ^ yd o f. When V is of finite
dimension we can identify V and its bidual (Vd)d, in which case we have
that (/*)* = / . Moreover, if / : V -> W is represented relative to fixed
ordered bases by the matrix A then /* : Wd —> Vd is represented relative
to the corresponding dual bases by the transpose A* of A.
If V is a finite-dimensional inner product space then the mapping
^ i i H ^ describes a conjugate isomorphism from V to Vd, by which
we mean that
(x + y)d = xd + yd and (Xx)d = Xxd.
The adjoint f* :W -+V of f :V ->W is defined by
and is the unique linear transformation such that
(Vx,yeV) (f(x)\y) = (x\f*(y)).
We say that / is normal if it commutes with its adjoint. If the matrix
of / relative to a given ordered basis is A then that of /* is A . We say
that A is normal if it commutes with A . A matrix is normal if and only
if it is unitarily similar to a diagonal matrix, i.e. if there is a matrix U
with U~l = U such that U~1AU is diagonal. A particularly important
type of normal transformation occurs when the vector space in question
is a real inner product space, and topics dealt with in this section reach
as far as the orthogonal reduction of real symmetric matrices and its
application to finding the rank and signature of quadratic forms.
19
The dual of a vector space V over a field F is the vector space Vd =
£(V,F) of linear functional f : V -+ F. If V is of finite dimension
and B = {v\,..., vn} is a basis of V then the basis that is dual to B is
Bd = {vf,..., v*} where each vf : V -» F is given by
For every x G V we have
a? = vf(x)vi +
and for every / £ Vd we have
If (v,) n , (^t)n are ordered bases of V and (r^) n , (wf )n the correspond-
ing dual bases then the transition matrix from (vf ) n to (wf ) n is (-P"1)*
where P is the transition matrix from (v,) n to (ty,) n . In particular,
consider V = \Rn. Note that if
is a basis of IRn then the transition matrix from B to the canonical basis
(c,*)n of \Rn is M = [wiy]ftXn where mt-y = ayt-. The transition matrix
from J5d to (e^)n is given by (M~1)t. We can therefore usefully denote
the dual basis by
, 2: Duality and normal transformations
where [an,..., a t n ] denotes the ith row of M~l, so that
• • - + ainxn.
The fadua/ of an element xisxA :Vd -> F where z A (t/ d ) = yd(x). It
is common practice to write yd{x) as (x, yd) and say that j / d annihilates
x if (z, t/d) = 0. For every subspace W of V the set
= { / G 7 d | (Vx G W) (z, / ) = 0}
is a subspace of W and
1
=dim7.
The transpose of a linear transformation f : V —> W is the linear
mapping /* : Wd -* V d described by yd ^ yd o f. When V is of finite
dimension we can identify V and its bidual (Vd)d, in which case we have
that (/*)* = / . Moreover, if / : V -> W is represented relative to fixed
ordered bases by the matrix A then /* : Wd —> Vd is represented relative
to the corresponding dual bases by the transpose A* of A.
If V is a finite-dimensional inner product space then the mapping
^ i i H ^ describes a conjugate isomorphism from V to Vd, by which
we mean that
(x + y)d = xd + yd and (Xx)d = Xxd.
The adjoint f* :W -+V of f :V ->W is defined by
and is the unique linear transformation such that
(Vx,yeV) (f(x)\y) = (x\f*(y)).
We say that / is normal if it commutes with its adjoint. If the matrix
of / relative to a given ordered basis is A then that of /* is A . We say
that A is normal if it commutes with A . A matrix is normal if and only
if it is unitarily similar to a diagonal matrix, i.e. if there is a matrix U
with U~l = U such that U~1AU is diagonal. A particularly important
type of normal transformation occurs when the vector space in question
is a real inner product space, and topics dealt with in this section reach
as far as the orthogonal reduction of real symmetric matrices and its
application to finding the rank and signature of quadratic forms.
19
