A problem from the tensor section in my compendium states that $LL^t=I \implies L^tL=I$ needs to be shown.
what I did was $(LL^t)_{ij} = L_{ik}L^t_{kj}=L_{ik}L_{jk}=\delta_{ij}$, and $(L^tL)_{ij} = L^t_{ik}L_{kj}=L_{ki}L_{kj}=\delta_{ij}$.
My first question is how can one show, or why does $L_{ik}L_{jk}=\delta_{ij}$? The second question is, is this $L_{ki}L_{kj}=\delta_{ij}$ true? (Maybe the answer to the second question gets answered by the first). The third question is does what I've shown really answer the original question? I don't see that what I have done implies what they want, maybe an equivalence?