Basic Linear Algebra - Computer Graphics and Geometric Modeling - page 879

so that

( )( ) =ƒ ( ) ( )

TT ¥ uu , T T u ƒ u .

1 2 1 2 1 2 1 2

Because we do not have to worry about parenthesizing tensor products by Theorem
C.6.2(3), this construction generalizes to produce a unique linear transformation

TT ƒ ƒ◊◊◊ƒ k : UU ƒ ƒ◊◊◊ƒ U VV Æ ƒ ƒ◊◊◊ƒ V .

1 2 1 2 k 1 2 k

Definition. The map T 1 ƒ T 2 ƒ ···ƒ T k is called the tensor product of the maps T i .

Now, although it is clear from the definition that tensor products have to do with
multilinear maps, we want to describe a much more fundamental relationship
between the two. See Theorem C.6.6 below. First of all we shall define a parallel
algebra structure on multilinear maps.

If f Œ L r ( V ) and g Œ L s ( V ), then define f ƒ g Œ L r+s ( V ) by

Definition.

( )( ) = ( ) ( ) fg ƒ vv , ,..., vv , ,..., v f vv , ,..., v g v ,..., v , (C.7)

12 r r + 1 rs + 12 r r + 1 rs +

for v i Œ V . The map f ƒ g is called the tensor product of f and g.

Let L( V ) denote the direct sum of the vector spaces L k ( V ), k ≥ 0.

Notation.

C.6.3. Theorem. The tensor product operation ƒ defined by equation (C.7) turns
L( V ) into an algebra called the algebra of real-valued multilinear maps on V k .

Proof. This theorem is the analog of Theorem C.6.2(1-3). One can easily prove that
ƒ is associative and that the distributive laws hold.

C.6.4. Theorem. There is a unique vector space isomorphism

() Æ (

)

k k

y : L V T V *, (C.8a)

with the property that

( ) = () ( )

g vv , ,..., v y g v ƒ v ƒ◊◊◊ƒ v (C.8b)

12 k 1 2 k

for all g ΠL k ( V ) and v i ΠV .

Proof. This is an easy consequence of the universal factorization property of the
tensor product.

C.6.5. Theorem. Let U and V be vector spaces.

(1) There is a unique isomorphism

( ) j : UV UV * ƒÆƒ * *