Basic Linear Algebra - Computer Graphics and Geometric Modeling - page 874

*

() =d vv .

i j ij

The map from V to V * that sends v i to v i * is a vector space

C.5.2. Theorem.

isomorphism.

Proof. Straightforward.

The isomorphism in Theorem C.5.2 between V and V * clearly depends on the
basis.

The basis v * , v * ,..., v * is called the dual basis of v 1 , v 2 ,..., v n . Definition.

Notice that if

n

 a ii
i

wv = ,

= 1

then v i *( w ) = a i , so that the ith dual basis element just picks out the ith component
coefficient of the expansion of a vector w in terms of the v i 's.

C.5.3. Theorem. Let V and W be vector spaces over a field k and let T : V Æ W be
a linear transformation. The map

T* : WV * Æ *

defined by

( () = ( () ) T * a v a T v , for v ΠV ,

is a linear transformation.

Proof. Easy.

Definition. The map T* in Theorem C.5.3 is called the dual map of the linear trans-

formation T.

Next, given v ΠV , define v ** ΠV ** = ( V *)* by

() = () v ** aa v , for a ΠV *.

C.5.4. Theorem. The map from V to V ** that sends v to v ** is a vector space iso-
morphism called the natural isomorphism between V and V **.

Proof. Easy.

Note that, although the isomorphism between V and V * depended on the choice
of a basis for V , the isomorphism between V and V ** does not. This allows us to iden-
tify V ** with V in a natural way and one often makes this identification.