Are the indices in $\Sigma_j A_{ji}e_j$ correct? A lot of times when I see a matrix-vector multiplication written in index form, it seems like the dimensions of the matrix are inverted or that the index they are using for the vector is the wrong one. I have seen this in many courses, but its never clear.
In my lecture notes for instance as part of a larger proof, they start like this:

Let $\{e_1,e_2,...e_n \}$ be an orthonormal basis of $V$ and $f:V\mapsto V $
an endormorphism. Let $A$ be the matrix representing the
endomorphism in this basis.
The columns of the matrix are: $f(e_i)=\Sigma_j A_{ji}e_j$

They didn't say it explicitly but I am pretty sure they are using column vectors as it is standard in linear algebra. So if the bases have $n$ elements I expect the matrix to be a $n\times n$ matrix, and therefore if the matrix $A$ is indexed as $A_{ji}$, $e$ should be indexed with $i=i\times1$ and not with $j$ as they are doing, that is I expect
$$f(e_i)=\sum_i A_{ji}e_i$$ or $$f(e_j)=\sum_j A_{ij}e_j$$
I don't think is a typo since they use it everywhere.
Could it be that I am unaware of some common usage? Could it be that they just don't care about the vector being a row or a column all the time and thus when they  write $\Sigma_j A_{ji}e_k$ with either $k=i$ or $k=j$   the vector $e_k$ is supposed  to be whatever it needs to be to make the expresion meaningful ( that is a column vector if they are using $j$ or  a row vector if they are using $i$) ?
Note: I found this hand-out: https://sites.math.washington.edu/~lee/Courses/443-2013/bilinear.pdfwhich is actually about the bilinear forms which is the context of my question and they do the same again, right from the first equation
$$A\mathbf{x}_j=\sum_{i=1}^nA_{ij}\mathbf{x}_j$$
 A: The notation is correct. In the equation
$$
Ax_j=\sum_{i=1}^n A_{ij}x_i\tag1
$$
the $i$ is an index of summation, whereas $j$ is "unbound". When the summation on the RHS is performed, the index $i$ disappears and the result (the RHS) involves $j$ only, same as the LHS.  If the index of summation in (1) were $j$ instead of $i$, the RHS would depend on $i$ after the dust settles, whereas the LHS would depend on $j$, which wouldn't make sense. (In fact if the index of summation in (1) were $j$ instead of $i$, we'd be able to factor out $x_i$ from the sum.)
In other words, the specific $x_j$ on the LHS is not the same as the generic $x_i$ on the RHS. In the RHS as $i$ ranges from $1$ to $n$ we visit all vectors $x_1,\ldots,x_n$, whereas in the LHS we stay with the specific vector $x_j$.
You should interpret (1) as a statement that holds for each $j$, i.e., it's an identity in $j$.
A: This is not a "literal" answer, but a sort of "meta-answer".
These indexing issues with matrices are an ever-recurring source of trouble.
Part of the problem is that these indexing things are not truly intrinsic, but only some sort of fallout from notational choices and conventions about the language involved.
Some decades ago, I did think there'd be some underlying ground about order of indices... but, after working hard to get down to it, found that the true answer is/was that the problem is not about mathematics, not even quite about "normal language", but about very volatile conventions of notation.
So, my serious advice to you is to re-appraise each situation...
