A 2020 Vision of Linear Algebra
So I like this outline. It is very MIT-ish where there is a sense of teaching someone to solve practical engineering problems with matrices.<p>But, I do foresee some difficulties. One thing that I find really difficult, for example, is that I take undergrads who have had linear algebra and ask &quot;what is the determinant?&quot; and seldom get back the &quot;best&quot; conceptual answer, &quot;the determinant is the product of the eigenvalues.&quot; Like, this is math, the best answer should not be the only one, but it should be ideally the most popular.  We would consider it a failure in my mind if the most popular explanation of the fundamental theorem of calculus was not some variation of &quot;integrals undo derivatives and vice versa&quot;. I don&#x27;t see this approach solving that. Furthermore there is a lot of focus from day one on this CR decomposition which serves to say that a linear transform from R^m to R^n might map to a subspace of R^n with smaller dimension r &lt; min(m,n) and while in some sense this is true it is itself quite &quot;unphysical&quot;—if a matrix contains noisy entries then it will generally only be degenerate in this way with probability zero. (You need perfect noise cancelation to get degeneracies, which amounts to a sort of neglected underlying conserved quantity which is pushing back on you and demanding to be conserved.) In that sense the CR decomposition is kind of pointless and is just working around some &quot;perfect little counterexamples&quot;. So it seems weird to see someone say &quot;hold this up as the most important thing!!&quot;