I just looked up my old class: linear algebra basics; Gaussian/LU/Choleski decomp; determinants; normed spaces; condition number; iterative methods; Euclidean spaces; QR decomp; Hermitian geometry; Eigenscheissen; spectral theorems; finite elements method; SVD and pseudoinverses; quadratic optimization.
Can confirm, totally useless in the non-research world. The most applicable task was computing spline curves. Even the Google Images result for "hermitian geometry" is mostly images of research papers. How is that real-world relevant?
The useful robotics stuff for me is either already written as a library I can call or pseudocode I can find in AIMA. Then again, all of that stuff had to come from somewhere and receive the optimization treatment.
I'd liken that entire Michigan mathematics course to the first week of my FEA course. "Here's how to calculate, by hand, a basic example of stress and strain in a very simple geometry using matrix operations. Cool, now that you see how much of a real hassle that is, never do it again because we have Ansys and Solidworks."
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On the other hand, a paper such as http://ras.papercept.net/images/temp/IROS/files/3131.pdf would seem totally inaccessible without a class like this Robotics 501. Maybe that constains good examples of the math being instructed. (Disclaimer: I didn't do more than glance at the course material and watch a few moments from the lectures.)
For instance, you'd like to know that when a state-space matrix is not full rank, you lose some control. In practical terms, a robot arm can usually move in any arbitrary Cartesian direction. A fully outstretched robot arm cannot.
For computing inverse kinematics... It's much simpler for a processor to have a slow update cycle that computes how every motor's changes will affect the position of the end effector (robot "tip") and then take a numerical inverse and use that to figure out a time-local, space-local approximation of getting to your eventual goal. It's a helluva lot easier than solving exactly.
I also recall some of the mathematics being relevant to collision detection (both in planning and execution stages), but the details are hazy. Basically, computation is expensive and convex hulls and precomputation can save a lot of cycles until it matters.
Yet other linear algebra lands in the realm of computer vision, optimization, and finite-element analysis.