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First, there's the idea of how to use the superscript and subscript in a transformation/vector. These "index gymnastics" are quite mindboggling to me.

For example, in our notes, the transformed basic vectors [itex]\hat{e'}_{i}=\sum_{j=1}^{N}{e}_{j}A_{j}^{i}[/itex] with A_{j}^{i} being the transformation matrix. The placement of the j's and i's baffle me, and I feel as if I don't get this figured out, I'll be completely lost from here on out (Right now one of our homework problems involves creating these transformation matrices to an oblique system).

See this link for the lecture material (you'll find slides 1-5 describe what I'm having trouble with).