Before working with tensors, it is helpful to introduce a new notation for working with the standard vectors and matrices familiar in linear algebra. In linear algebra, and multivariable calculus, operations with vectors and matrices are typically expressed by some symbol where the operation is implied. For example, the dot product between two vectors u and v is expressed as
\mathbf{u} \cdot \mathbf{v} =
\begin{pmatrix} u_1 & u_2 & \cdots \end{pmatrix}
\begin{pmatrix} v_1 \\ v_2 \\ \vdots \end{pmatrix}
Where the actual operation of dotting the vectors is implied. In the new index notation, the operation will be described explicitly in terms of the components of each vector. In the case of the dot product, then we will write
\mathbf{u} \cdot \mathbf{v} = \sum_i u_i v_i
where u_i is the i\mathrm{th} element of u.
Similarly, whereas in linear algebra one would write the operation of a matrix A on a vector \mathbf{w}, \mathbf{w} = A\mathbf{v} as
\begin{pmatrix} w_1 \\ w_2 \\ \vdots \end{pmatrix} =
\begin{pmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
\end{pmatrix}
\begin{pmatrix} v_1 \\ v_2 \\ \vdots \end{pmatrix}
In the index notation, this operation is expressed as
w_i = \sum_j a_{ij} v_j
which can be read as "the i\mathrm{th} element of w is equal to the dot product of j with the i\mathrm{th} row of A."
One nice thing about the new notation is that since everything is written in terms of components, which are just numbers, everything commutes. For example, the product of two matrices A and B is written in index notation as
c_{ij} &= \sum_k a_{ik} b_{kj} &= \sum_k b_{kj} a_{ik}
Note
Of course, the fundamental composition of transformations which AB represents still does not commute in general, i.e.
\sum_k a_{ik} b_{kj} &\neq \sum_k a_{kj} b_{ik}
If you look back, you'll notice that in every case where we have a repeated index in a product, there is always a sum over that index. The Einstein convention is to just drop writing the sum explicitly, and assume that every time a repeated index appears in a product, the sum is implied, i.e. we can write
c &= \mathbf{u} \cdot \mathbf{v} \qquad &\Leftrightarrow \qquad c &= u_i v_i \\
w &= A \mathbf{v} \qquad &\Leftrightarrow \qquad w_i &= a_{ij} v_j \\
C &= A B \qquad &\Leftrightarrow \qquad c_{ij} &= a_{ik} b_{kj}