System of equations and Einstein notation I'm confused about writing a system of equations in Matrix form and Einstein notation
Let's start with what I understand:
I have the following system of n equations (let's take n=2 to make it easier to write):
$$a_{11}u_1+a_{12}u_2=b_1$$
$$a_{21}u_1+a_{22}u_2=b_2$$
I can write this in matrix form as:
$$
    \begin{bmatrix}
    a_{11} & a_{12}\\\
    a_{21} & a_{22}
    \end{bmatrix}
\begin{bmatrix}
    u_1 \\\
    u_2 
    \end{bmatrix}=
\begin{bmatrix}
    b_1 \\\
    b_2 
    \end{bmatrix}
$$ 
Or using the implied summation: $a_{ji}u_i=b_j$.
Now, where I get confused is that I need to multiply the left hand side by $c_kx_k$, so my system of equation is now (u and x are two input vectors, $a_{ji}$, and $c_k$ are constant values, and $b_j$ is the RHS result):
$$(c_1x_1+c_2x_2)(a_{11}u_1+a_{12}u_2)=c_1x_1a_{11}u_1+c_1x_1a_{12}u_2+c_2x_2a_{11}u_1+c_2x_2a_{12}u_2=b_1$$
$$(c_1x_1+c_2x_2)(a_{21}u_1+a_{22}u_2)=c_1x_1a_{21}u_1+c_1x_1a_{22}u_2+c_2x_2a_{21}u_1+c_2x_2a_{22}u_2=b_2$$
Can I write this system of equations using Einstein notation as follows: $c_kx_ka_{ji}u_i=b_j$?
And how would I write this in a Matrix notation? I have the feeling that the parameters a and c can be combined in a 2x2x2 matrix, with the first "slice" being $c_1a_{ji}$ and the second slice $c_2a_{ji}$. But then I do not see how this matrix can be multiplied by the input vectors x and u, with are both 1x2 vectors.
Thanks for the help.

Your Einstein notation is correct. Looking at the indices, you can see that $\mathbf{x}$ and $\mathbf{c}$ are both contracted over the same index. In terms of matrix-vector notation, you can write this as $\mathbf{c}^{\text{T}}\mathbf{x}$. 
Note that after performing this operation, the result is just a number, call it $k$. You can see from your Einstein notation that none of the other quantities depend on the same indices. So your modified matrix equation could really just be written as $k\mathbf{A}\mathbf{u}=\mathbf{b}$
