Linear Maps Between Vector Spaces: Higher Order Tensors
First and Second Order Tensors in 
For the discussion in this section, we will assume a right handed orthonormal basis set and an alternate right handed orthonormal basis set
. The two basis sets are related by the matrix
whose components
.
If has components
and
in the basis set set
, and
is the representation of
in the basis set
with components
and
, then, these components are related with the relationship:
(1)









(2)



Third Order Tensors in 
The linear map where
is the set of all linear operators
is called a third order tensor.
In the following we will show how the components of change when the orthonormal basis set for the underlying space
is changed.
The matrix representation of when
is chosen as the basis set for
has
independent components
such that
such that:
(3)




(4)


(5)

(6)


(7)


Fourth Order Tensors in 
The linear map where
is the set of all linear operators
is called a fourth order tensor.
In the following we will show how the components of change when the orthonormal basis set for the underlying space
is changed.
The matrix representation of when
is chosen as the basis set for
has
independent components
such that
such that:
(8)




(9)




(10)

(11)


(12)

