So how exactly is the Einstein lower and upper index convention?
Wikipedia says that lower indices refer to linear functionals in dual spaces of vectors and upper indices to vectors themselves.
E.g. the inner product is written as:
$$u \cdot v = u_j v^j$$
However then some of the examples seem to write this the other way around:
Outer product:
$$A^i_j = u^i v_j$$
or matrix-vector multiplication:
$$A^i_j v^j$$
This would mean that $v_j$ is linear functional and $u^i$ acts as a coordinate.
However, now $v_j$ has no parameters? So how can it be a functional?
Or perhaps this has something to do with the Einstein convention not caring about order in which is written? That $u_j v^j = v^j u_j$?
Bear in mind that, while $u_i,\,u^i$ aren't the same thing, there's a simple link between them. (Maybe I should say $u_i$ has a simple link to $u^j$, because the link requires all values of the index on each expression to be considered; in general there isn't, for example, a nice relation of $u_1$ to $u^1$, although you can compute $\frac{\partial u_i}{\partial u^j}$.) Explicitly $u_i=g_{ij}u^j,\,u^j=g^{jk}u_k$, with $g_{ij}$ the index-lowering metric tensor (often just the metric tensor) and $g^{jk}$ the index-raising metric tensor. These tensors are symmetric, and clearly $g_{ij}g^{jk}=\delta_i^k$.
We can therefore rewrite our equations so any given vector/linear functional always has its indices at the same height. For example, I could write $u\cdot v=g^{jk}u_j v_k,\,A^i_j=g^{ik}u_k v_j,\,A^i_j v^j=A^i_j g^{jk}v_k$. In those equations, I give the vectors subscripts (well, I suppose you'd call them linear functionals); I encourage you to work out what happens if they're given superscripts instead.
Bear in mind "the linear functionals on this vector space" are themselves the members of another vector space (you can verify the set of such functionals satisfies the usual axioms). You can, if you prefer, think entirely in terms of the contents of one vector space. But if you want to think about both of them, bear in mind that all calculations give the same results as the one-space approach with appropriate metric tensors kicking round. Any "inconsistency" you've detected boils down to which sources want to emphasize the two-space conceptualisation.