6
1 Some Linear Algebra
Definition 1.4.1 Let U and V be two finite-dimensional vector spaces over the same
field K . The tensor product of U and V is a vector space U ⊗ V over K together with
a bilinear function B U ⊗V : U × V −→ U ⊗ V written as B(u, v) = u ⊗ v, with
the universal property: for every vector space W over K and any bilinear function
B : U × V −→ W , there exists a unique linear transformation T w : U ⊗ V −→ W
such that T w ◦ B = B. The elements u ⊗ v are said to be tensor of u and v.
Note that in Definition 1.4.1 we have written “The tensor product of U and V ...”;
this can be done because tensor products are unique up to isomorphisms, according
to the next theorem.
Theorem 1.4.1 If (T 1 , B T 1 ) and (T 2 , B T 2 ) are tensor products of the vector spaces
U and V , then there exists a unique isomorphism f : T 1 −→ T 2 such that f ◦ B T 1 =
B T 2 .
Exercise 1.4.1 Show the uniqueness of tensor products up to isomorphisms.
If the vector spaces are finite-dimensional then the tensor product between them
exists.
Theorem 1.4.2 Let U and V be finite-dimensional vector spaces over K with
dimensions m and n, respectively. Then U ⊗ V exists: if {u 1 , u 2 , . . . , u m } is a basis
for U and {v 1 , v 2 , . . . , v n } is a basis for V then the mn vectors u i ⊗ v j for all
i = 1, 2, . . . , m and j = 1, 2, . . . , n, form a basis for U ⊗ V , i.e., dim(U ⊗ V ) =
[dim(U )][dim(V )].
Exercise 1.4.2 Show the existence of tensor products.
Remark 1.4.1 (1) In Definition 1.4.1, we have defined the tensor product of two
vector spaces. The generalization for a finite number of vector spaces is performed
in the same way by applying the concept of multi-linear functions. More precisely,
if W and V 1 , V 2 , . . . , V n are vector spaces over the same field K , a function M :
V 1 × V 2 × · · · × V n −→ W is said to be multi-linear if, fixing n − 1 variables (except
the ith variable), M is a linear transformation in the ith variable. Thus, we can also
consider a finite number of tensor products of vectors spaces V 1 ⊗ V 2 ⊗ · · · ⊗ V n .
As a consequence of its definition, the tensor product satisfies the properties given
below:
(i) ∀ z ∈ K , v ∈ U and w ∈ V one has z(v ⊗ w) = (zv) ⊗ w = v ⊗ (zw);
(ii) ∀ v 1 , v 2 ∈ U and w ∈ V , (v 1 + v 2 ) ⊗ w = (v 1 ⊗ w) + (v 2 ⊗ w);
(iii) ∀ v ∈ U and w 1 , w 2 ∈ V , v ⊗ (w 1 + w 2 ) = (v ⊗ w 1 ) + (v ⊗ w 2 ).
1 Some Linear Algebra
Definition 1.4.1 Let U and V be two finite-dimensional vector spaces over the same
field K . The tensor product of U and V is a vector space U ⊗ V over K together with
a bilinear function B U ⊗V : U × V −→ U ⊗ V written as B(u, v) = u ⊗ v, with
the universal property: for every vector space W over K and any bilinear function
B : U × V −→ W , there exists a unique linear transformation T w : U ⊗ V −→ W
such that T w ◦ B = B. The elements u ⊗ v are said to be tensor of u and v.
Note that in Definition 1.4.1 we have written “The tensor product of U and V ...”;
this can be done because tensor products are unique up to isomorphisms, according
to the next theorem.
Theorem 1.4.1 If (T 1 , B T 1 ) and (T 2 , B T 2 ) are tensor products of the vector spaces
U and V , then there exists a unique isomorphism f : T 1 −→ T 2 such that f ◦ B T 1 =
B T 2 .
Exercise 1.4.1 Show the uniqueness of tensor products up to isomorphisms.
If the vector spaces are finite-dimensional then the tensor product between them
exists.
Theorem 1.4.2 Let U and V be finite-dimensional vector spaces over K with
dimensions m and n, respectively. Then U ⊗ V exists: if {u 1 , u 2 , . . . , u m } is a basis
for U and {v 1 , v 2 , . . . , v n } is a basis for V then the mn vectors u i ⊗ v j for all
i = 1, 2, . . . , m and j = 1, 2, . . . , n, form a basis for U ⊗ V , i.e., dim(U ⊗ V ) =
[dim(U )][dim(V )].
Exercise 1.4.2 Show the existence of tensor products.
Remark 1.4.1 (1) In Definition 1.4.1, we have defined the tensor product of two
vector spaces. The generalization for a finite number of vector spaces is performed
in the same way by applying the concept of multi-linear functions. More precisely,
if W and V 1 , V 2 , . . . , V n are vector spaces over the same field K , a function M :
V 1 × V 2 × · · · × V n −→ W is said to be multi-linear if, fixing n − 1 variables (except
the ith variable), M is a linear transformation in the ith variable. Thus, we can also
consider a finite number of tensor products of vectors spaces V 1 ⊗ V 2 ⊗ · · · ⊗ V n .
As a consequence of its definition, the tensor product satisfies the properties given
below:
(i) ∀ z ∈ K , v ∈ U and w ∈ V one has z(v ⊗ w) = (zv) ⊗ w = v ⊗ (zw);
(ii) ∀ v 1 , v 2 ∈ U and w ∈ V , (v 1 + v 2 ) ⊗ w = (v 1 ⊗ w) + (v 2 ⊗ w);
(iii) ∀ v ∈ U and w 1 , w 2 ∈ V , v ⊗ (w 1 + w 2 ) = (v ⊗ w 1 ) + (v ⊗ w 2 ).