Lisantiom

2022-09-06

After studying general a linear algebra course, how would an advanced linear algebra course differ from the general course?
And would an advanced linear algebra course be taught in graduate schools?

At my [undergraduate] university [which was University of Cincinnati, at the time of this post], the first linear algebra sequence is taught to sophomores. It is mostly computational. Everything takes place in the reals and complex numbers. The class begins with row reducing and culminates with finding determinants and eigenvalues. I don't remember which book we use for this but it's terrible and the class is very easy.
Later, students are encouraged to take "abstract" linear algebra, which focuses on abstract vector spaces (though they are all assumed to be over fields of characteristic 0), inner product spaces, quadratic forms, proving the spectral theorem, and culminates with Jordan canonical form and the theory of convex sets. For this we use Lang's linear algebra. More emphasis is placed on the spectral theorem than anything, with Jordan form and convex sets only if the class moves fast enough so there's time.
Finally, after a student has taken the senior level abstract algebra sequence (featuring the basics of groups, rings, and fields), he may elect to take the graduate algebraic structures class, in which module theory, more advanced ring theory, and some representation theory are covered. For this class we use Dummit and Foote (and whichever other books we feel like). [At my incumbent university, University of Florida, basic ring and module theory is done in the first year graduate course, also using Dummit and Foote. Multilinear algebra (tensors) and more advanced ring theory are covered during spring semester of second year graduate algebra, which concurrently uses Lang, Hungerford, and Matsumura.]

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