It's finals week, so not much research. But I did get in a useful conversation with Soledad Villar (NYU) about the definition of a tensor (as opposed to a matrix, say). She had a different starting point than me! But we converged to similar things. You can think of it in coordinate-free or coordinate-transform ways, you can think of it in operator terms, and you can think of it as being composed of sums of outer products of vectors. I always thought of a tensor as being a ratio of vectors, in some sense, but that's a very hand-wavey idea.
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