##
Math 55a: Duality basics

The *dual* of an *F*-vector space *V*
is the *F*-vector space Hom(*V*,*F*)
of linear maps from *V* to *F*,
often called *functionals* on *V*.
We'll usually denote the dual of *V* by *V*^{*}.
If *W* is the direct sum of *U* and *V*
Then *W*^{*} is the direct sum
of *U*^{*} and *V*^{*};
for instance, the copy of *U*^{*}
in *W*^{*} consists of the functionals
mapping all vectors in *U* to zero
(the ``annihilator'' of *U* in *W*^{*}).
The dual of a quotient space *V/U*
is naturally a subspace of *V*^{*},
namely the annihilator of *U* in *V*^{*}.
If *V* has finite dimension *n*
then so does *V*^{*}.
The dimension of the annihilator of *U*
in *V*^{*}
then equals the *co*dimension of *U* in *V*.
If we choose a basis of *V*,
and use it to identify elements of *V* with ``column vectors''
of length *n*, then elements of *V*^{*}
correspond to ``row vectors'' of the same length.
There is in general no canonical isomorphism between
*V* and *V*^{*},
even if *V* is finite dimensional.
We do, however, have a canonical map from *V*
to its *second dual* *V*^{**},
which is an isomorphism when *V* is finite dimensional.
The key idea is that *v*^{*}(*v*)
is ``bilinear'': not only is it a linear function of *v*
for each choice of *v*^{*}, but it's also
a linear function of *v*^{*} for each choice
of *v* -- so each *v* may be considered as
a functional on *V*^{*},
that is, as an element of *V*^{**}!

If *T* is a linear transformation from *V* to *W*
then we get a linear transformation *T*^{*}
from *W*^{*} to *V*^{*},
called the *adjoint* of *T* and denoted by
*T*^{*}, defined as follows:
*T*^{*}(*w*^{*})
is the element of *V*^{*} taking any *v* to
*w*^{*}(*T*(*v*)).
The adjoint is itself a linear map from
Hom(*V*,*W*) to
Hom(*W*^{*},*V*^{*}).
This map is an isomorphism when *V* and *W*
are finite dimensional. In that case, if we choose bases
for *V* and *W* then the matrix of *T*^{*}
with respect to the dual bases
is the *transpose*
of the matrix of *T* with respect to our bases for *V,W*.
Naturally *T*^{**} is identified with *T*
under the canonical maps that identify
*V*^{**} and *W*^{**}
with *V* and *W*.