Page 402 - Advanced Linear Algebra

386 Advanced Linear Algebra

i

i
; ²= ³ ~ = nÄn= n = nÄn= i ~ = n n²= ³ n
’••••“••••”
’•••“•••”

factors factors

is called the space of tensors of type ² Á ³ , where is the contravariant type
and is the covariant type . If ~ ~ , then ; ²= ³ ~- , the base field. Here

we use the notation = n for the -fold tensor product of with itself. We will

=
also write = d for the -fold cartesian product of with itself.

=
Since =š = ii , we have
³ š hom
in
in
id

; ²= ³ ~ = n n ²= ³ š ²²= ³ n = n i - ²²= ³ d = d Á -³

which is the space of all multilinear functionals on
i
i
’••••“••••” d = dÄd=
= d Äd=
’•••“•••”
factors factors
In fact, tensors of type ² Á ³ are often defined as multilinear functionals in this
way.
Note that

dim²; ²= ³³ ~ ´ dim²= ³µ b

Also, the associativity and commutativity of tensor products allows us to write

b
; ²=³ n ; ²=³ ~ ; b ²=³

at least up to isomorphism.
Tensors of type ² Á ³ are called contravariant tensors

;²= ³ ~ ;²= ³ ~ = n Ä n =
’•••“•••”

factors
and tensors of type ² Á ³ are called covariant tensors
i

;²= ³ ~ ; ²= ³ ~ = n Ä n = i
’••••“••••”

factors
Tensors with both contravariant and covariant indices are called mixed tensors .
In general, a tensor can be interpreted in a variety of ways as a multilinear map
on a cartesian product, or a linear map on a tensor product. Indeed, the
interpretation we mentioned above that is sometimes used as the definition is
only one possibility. We simply need to decide how many of the contravariant
factors and how many of the covariant factors should be “active participants”
and how many should be “passive participants.”