Those references to "group characters" and "representation theory" cracked me up, in a good way. For those who have an engineer's exposure to Fourier series and

Those references to "group characters" and "representation theory" cracked me up, in a good way. For those who have an engineer's exposure to Fourier series and transforms, these would be the logical next steps to appreciate the big picture. As a pedagogic device I get some success with is by keeping things concrete, showing that shift in time (index) of a very high dimensional vector, is same as multiplying with a fixed matrix. Same with the difference operator, summation operators and so on. Then one can think of singular value, or eigen-decomposition of those matrices, that's the point were students realize why these operators "become" point-wise multiplications and divisions. Works with those who have some experience with finite dimensional matrices.