# 111 Actions

2018 2017 2016 Jul 9 answered How to intuitively convince the students that a strip with two full twists is homeomorphic to the standard annulus? Apr 29 awarded Caucus Mar 13 awarded Yearling Dec 9 revised Word for the dimension of the vector space in which a vector lives? added 268 characters in body Dec 9 asked Word for the dimension of the vector space in which a vector lives? Oct 30 awarded Nice Answer Mar 13 awarded Yearling Jul 14 revised Teaching strong induction instead of induction added 11 characters in body May 18 comment Do “gateway tests” work? I teach at Michigan, including I taught Applied Linear Algebra which instituted a Gateway on row reduction and other matrix operations while I was there. My post-gateway syllabus has an extra week of material in it, because I spend so much less time on these basics. But I don't have the sort of formal data you want. Mar 13 awarded Yearling Jan 20 revised Is there any proof of the fundamental theorem of algebra that can be introduced to undergraduates who have just completed Calc III? added 1 character in body Oct 24 awarded Nice Answer Jul 26 revised Is there any proof of the fundamental theorem of algebra that can be introduced to undergraduates who have just completed Calc III? edited body May 3 comment How to get students in a under-graduate linear algebra course interested in determinants? @LoopSpace How about: Given two $n \times n$ matrices $A$ and $B$, find all $t$ for which $t^2 \mathrm{Id} + t A + B$ is singular? For $t \mathrm{Id} + A$, you can use specialized methods for computing eigenvalues but, as far as I know, you need to fall back on determinants once the polynomial is higher degree in $t$. Mar 24 revised How can one motivate the adjugate matrix? added 20 characters in body Mar 24 answered How can one motivate the adjugate matrix? Mar 13 awarded Yearling Jan 21 comment How to motivate the surface element I wonder whether it might be useful to spell out the $k=2$ case: $\det \left( \begin{smallmatrix} \vec{u} \cdot \vec{u} & \vec{u} \cdot \vec{v} \\ \vec{v} \cdot \vec{u} & \vec{v} \cdot \vec{v} \end{smallmatrix} \right) = |u|^2 |v|^2 - |u|^2 |v|^2 \cos^2 \theta = |u|^2 |v|^2 \sin^2 \theta$. Sometimes I find that students whose eyes are glazing over in linear algebra wake up when I remind them of high school geometry/trig. (And sometimes I learn that they never had high school geometry, which is a different problem.) Jan 21 revised How to motivate the surface element deleted 2 characters in body Jan 21 comment How to motivate the surface element For someone who really understands QR decomposition, this is the same as Dirk's answer, but I claim it is more friendly to those who don't.