Related to HW8, the video below discusses eigenvectors for the matrix problems and also problem 2, the double dot quantum well. Please feel free to post any questions here. Also here is a page related to the problem 2 showing the basis states and one energy eigenstate. If you don't fully understand any part of problem 2, please ask about it. The linked paper in problem 2 is where our notation comes from.

Good question. Essentially, there is no difference. It is not one or the other, but both. The vector is a sequence of numbers. It can be written as a row or a column. It is essentially the same vector either way.
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