Linear Algebra: Fall 2012

MWF 10:10-11:10

Trout Hall Room 302

Announcements:

Exam #2 will take place in class on Friday 11/16

The Final Exam will take place at 10am on Wednesday 12/5 Strang's 4 fundamental subspaces:  the row space of A (orthogonal to the null space of A) and the column space of A (orthogonal to left null space of A)

Questions for further thinking:

(1) In class we briefly discussed how the set of real 2x2 matrices of determinant 1 (called the special linear group, SL(2,R)) "acted" on the upper half plane H={a+bi: b>0} through Mobius transformations.  Find the set of all matrices that leave the element i unchanged.  In other words, find the matrices A in SL(2,R) such that A*i=i.  The set of all such matrices is called the stabilizer of i, denoted Stab(i).

(2) Recall that a matrix group is a set of matrices that is closed under multiplication and under taking multiplicative inverses.  Is Stab(i) a matrix group?  Prove your assertion.

(3)  Does the multiplicative cancellation law hold for (square) matrices?  In other words, does AX=AB automatically imply that X=B?  Prove your assertion or find a counterexample.  (Hint:  it is probably helpful to test cases with 2x2 matrices to help you formulate a conjecture.)