I was watching this video: https://www.youtube.com/watch?v=RDkwklFGMfo
And the professor is talking about the inner product... then he brings up the "canonical" representation of the inner product in R^N.
What, specifically, does he mean? I see it a lot in matrix talk, but I'm never really sure what they mean.
It is not the usual terminology. "Canonical" has a well established use in linear algebra as a construction that is independent of a basis, or at least invariant with respect to different choices of basis. The thing in the lecture slides (around 25min in the video) is ordinarily called the "standard" inner product associated to a basis, or the "Euclidean" inner product.