Let \(\mathbf{T}\) be a second-order tensor (dyadic) and let \(\mathrm{V}\) be a vector. Then we have seen that \(\mathbf{T} \cdot \mathrm{V}\) is another vector, say U. Write the equations giving each of the three components of \(\mathrm{U}\) as linear combinations of the three components of \(\mathrm{V} ;\) write these equations in matrix form, dyadic form, and summation form. Observe that the nine coefficients in these linear combinations are just the nine components of \(T\).

Second-order tensors can interact with vectors to produce new vectors. This interaction is known as tensor-vector multiplication. For example, consider a tensor \(\mathbf{T}\) and a vector \(\mathbf{V}\). When we multiply these, we get another vector \(\mathbf{U}\).
In simple terms, imagine we have a set of rules (the tensor) that transforms the input vector into a new output vector. This operation follows defined mathematical properties.
Understanding tensor-vector multiplication is essential because it translates complex multidimensional data systems into tangible vector outputs, making it easier to interpret and analyze.

Tensors can be represented using matrices, a familiar format for many students. Let's break down how we represent the multiplication of a tensor \(\mathbf{T}\) with a vector \(\mathbf{V}\) in matrix form.
Suppose \(\mathbf{T}\) is represented by a 3x3 matrix and \(\mathbf{V}\) by a 3x1 column vector. The multiplication operation can be represented as:
\[ \begin{pmatrix} U_1 \ U_2 \ U_3 \end{pmatrix} = \begin{pmatrix} T_{11} & T_{12} & T_{13} \ T_{21} & T_{22} & T_{23} \ T_{31} & T_{32} & T_{33} \end{pmatrix} \begin{pmatrix} V_1 \ V_2 \ V_3 \end{pmatrix} \]
This matrix multiplication results in a new vector \(\mathbf{U}\), where each element is a linear combination of the components of vector \(\mathbf{V}\) weighted by the corresponding elements of matrix \(\mathbf{T}\). This form is helpful for computational purposes and visualizing the transformation.

In the dyadic form, the operation is expressed using outer products of basis vectors. This representation can be more abstract but shows a deeper insight into tensor operations.
The dyadic form of tensor-vector multiplication for tensor \(\mathbf{T}\) and vector \(\mathbf{V}\) can be written as:
\[ \mathbf{T} = \sum_{i=1}^{3} \sum_{j=1}^{3} T_{ij} \mathbf{e}_i \otimes \mathbf{e}_j \] where \(\mathbf{e}_i\) are basis vectors.
Thus, when multiplying with vector \(\mathbf{V}\), we get:
\[ \mathbf{U} = \mathbf{T} \cdot \mathbf{V} = \left( \sum_{i=1}^{3} \sum_{j=1}^{3} T_{ij} \mathbf{e}_i \otimes \mathbf{e}_j \right) \cdot \left( \sum_{k=1}^{3} V_k \mathbf{e}_k \right) \]
Using properties of the tensor product, this simplifies to:
\[ \mathbf{U} = \sum_{i=1}^{3} \sum_{j=1}^{3} T_{ij} V_j \mathbf{e}_i \]
The dyadic form highlights how the tensor components are coupled with the vector components to form the result, emphasizing the relationship between different dimensions.

Summation notation offers a concise way to express tensor operations. This notation is especially useful for generalizing the math to higher dimensions.
For tensor-vector multiplication, the components of the resulting vector \(\mathbf{U}\) are given in summation notation as:
\[ U_i = \sum_{j=1}^{3} T_{ij} V_j \]
Here, \(U_i\) represents the i-th component of the resulting vector \(\mathbf{U}\), \(T_{ij}\) are the components of the tensor, and \(V_j\) are the components of the vector \(\mathbf{V}\).
Summation notation simplifies the representation by removing the need for explicit looping constructs, making the equations more readable. It also prepares you for more advanced topics in linear algebra and tensor theory where such notation becomes indispensable.