Similar Matrices and Change of Basis (Algebra 2: Lecture 25 Video 1)

Lecture 25: We began this lecture by asking, 'When are two matrices are associated to the same linear transformation under different choices of bases?'  We defined what it means for two n x n matrices to be similar, and what is means for linear transformations to be similar.  We defined the transition matrix/change of basis matrix.  We examined the key properties of this matrix and saw why it gets this name.  We saw that two matrices associated to the same linear transformation with respect to two different bases are similar.  We also saw that give two similar n x n matrices we can construct a linear transformation and a pair of bases such that these matrices represent the same linear transformation with respect to different bases.  We discussed the connection between similarity classes and the conjugation action of GL(V) on Hom_F(V,V).  We then discussed an example.  In the last part of the lecture we discussed some additional things from Section 11.2 and from the Exercises following Section 11.2 that would be good to be familiar with (these will not be tested on the Final Exam).  We also discussed some highlights from Sections 11.3 on Dual Vector Spaces and from Section 11.4 on Determinants.

Reading: In the first part of this lecture we closely followed part of Section 11.2, pages 419-420.  We then discussed some exercises following Section 11.2.  We also gave a quick overview of some highlights of Sections 11.3 and 11.4.  I definitely recommend that you read the Exercises following Section 11.2 to get a better sense of what facts that you likely learned in a first linear algebra course focusing on matrices with entries in the real numbers carry over more generally to linear algebra over a general field F.  I recommend that you read Sections 11.3 and 11.4, but I do not recommend that you spend a lot of time and energy thinking about this material before the Final Exam.  Nothing on the Final Exam will require you to know anything from Sections 11.3 or 11.4.