Write equations (2,11), (2,12),(2,13),(2,14),(2,16),(2,17),(2,18) using summation convention.

Vector U and vector V with components $U_1,U_2,U_3$and $V_1,V_2,V_3$respectively.

The tensor equations use a lot of summation signs it would be a simplification. The summation convention omits the summation signs in equations and simply understand a summation over any index which appears exactly twice in one term.

Equation (2,11) can be written as follows.

${x_i}^{\text{'}}=a_{ij}{x}_j$

Equation (2.12) can be written as follows.

${V_i}^{\text{'}}=a_{ij}{V}_j$

Equation (2.13) can be written as follows.

$V_j=a_{ji}{V_j}^{\text{'}}$

Equation (2.14) can be written as follows.

${T_{kl}}^{\text{'}}=a_{ki}{a}_{lj}{T}_{ij}$

Equation (2.16) can be written as follows.

${U_i}^{\text{'}}=a_{ij}{U}_j$

Equation (2.17) can be written as follows.

${U_k}^{\text{'}}{V_l}^{\text{'}}=a_{ki}{a}_{lj}{U}_i{V}_j$

Equation (2.18) can be written as follows.

T${T_{\alpha \beta \gamma }}^{\text{'}}=a_{\alpha i}{a}_{\beta j}{a}_{\gamma j}{T}_{ijk}$

The formula for the cartesian vectors states that$V_i=\sum _{j=1}^3{a}_{ij}{V_j}^{\text{'}}$

$$\begin{gathered} V_i=\sum _{j=1}^3{a}_{ij}{V_j}^{\text{'}} \\ {V}_i\left(U_i\right)=\sum _{j=1}^3{a}_{ij}{V_j}^{\text{'}}\left({U_j}^{\text{'}}\right) \\ \sum _{id}{a}_{ki}{a}_{lj}{V_l}^{\text{'}}{U_k}^{\text{'}}=U_i{V}_j \end{gathered}$$

Hence The statement has been proven.