## 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 Q 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)

Similarly, consider the tensor . If its matrix representation has components when is chosen as the basis set for , and if is the matrix representation with components when is chosen as the basis set for , then, these components are related with the relationship:

(2)

Vectors in have independent components and are called first order tensors. Linear operators from have independent components and are termed second order tensors.

### 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)

Similarly, If is chosen as the basis set, then the components would relate the components of and as follows:

(4)

However, premultiplying (3) with and summing over produces:

(5)

By replacing the components of in (5) with the inverse of the relationship (2) and using (1) we get:

(6)

Comparing (4) and (6), the components are related to the components by the following relationship:

(7)

(7) could have also been obtained by noticing the following:

From which the relationship between the components of and can be obtained.

### 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)

Similarly, If is chosen as the basis set, then the components would relate the components of and as follows:

(9)

However, premultiplying (8) with the components and and summing over and results in:

(10)

By replacing the components of in (10) with the inverse of the relationship (2) and using 2 we get:

(11)

Comparing (11) and (9), the components are related to the components by the following relationship:

(12)

(12) could have also be obtained by noticing the following:

From which the relationship between the components of and can be obtained.