Obscure and weird change of indices

Question

I find the following argument with the indices to be extremely obscure-so obscure that I'm baffled. The following originate from my notes:

Where the basis $B =\left\{\vec{e}_{1},...,\vec{e}_{n}\right\}$ is Orthonormal, the linear operator L is defined by

$L\vec{e}_{j}=\sum_{k=1}^n L_{ji}\vec{e}_{j} $

which looks wrong. Free indices can be set to any 'value' as wished but it has to be consistent across all terms

Then, taking the inner product of $L\vec{e}_{j}$ and $\vec{e}_{i}$

$\langle\vec{e}_{i}$,$L\vec{e}_{j}\rangle =\langle\vec{e}_{i},\sum_{k=1}^nL_{kj}\vec{e}_{k}\rangle$

Why are the k indices there and more important the order of the row and column indices switched inconsistently?

Are this notes wrong?

Any input is appreciated. Thanks in advance.

Answer 1

Let $L$ be $2\times 2$ for simplicity. Then $$L = \begin{bmatrix} L_{11} & L_{12} \\ L_{21} & L_{22}\end{bmatrix} \\ \implies \sum_{k=1}^2L_{k1}\vec e_k = L_{11}\vec e_1 + L_{21}\vec e_2 = \begin{bmatrix} L_{11} \\ L_{21}\end{bmatrix} = \text{the first column of $L$} = L\vec e_1$$

Generalizing, we see that $$L\vec e_{j} = \sum_{k=1}^nL_{kj}\vec e_k$$ for any $n\times n$ matrix $L$.

So the first equation in your notes is just a typo. The second one with the inner product is correct, though.