General Information

As announced in the course syllabus, the second exam of the semester will be given in class on Friday, March 26. You will have the full class period to work on the exam. The format will be similar to that of the first exam -- four or five problems, each possibly containing several parts. One question again this time may consist of several ``true - false'' questions where you must either give a short proof (if you think the statement is true), or a counterexample (if you think the statement is false).

Topics to be Covered

This exam will cover all the material since the first exam, through and including the material on change of basis for matrix representations of linear maps from class on Monday, March 22 (Chapters 8 - 11 in the text). Of course, all of this depends heavily on the material on vector spaces, subspaces, bases, dimension, etc. from the first part of the course. You will need to have that material "under control" for this exam too. Specifically, the new topics for this exam are:

1. Linear mappings, kernels, images, etc.
2. The dimension theorem: If T : V -> W is linear and V is finite dimensional, then
   
   dim(V) = dim(Ker(T)) + dim(Im(T))
   

   and its consequences for injectivity, surjectivity, isomorphisms, etc.

3. Matrix representation of a linear mapping
4. Basic theorems about sums and compositions of linear mappings and the corresponding facts about matrices
5. Change of basis

Proofs to Know

1. Proof of the dimension theorem.
2. Let T : V -> W and S : W -> U be linear mappings between finite dimensional vector spaces. Then the composition ST : V -> U is linear. Moreover, if E,F,G are bases for V,W,U respectively, the matrix of T with respect to E,F is A, and the matrix of S with respect to F,G is B, then matrix of ST with respect to E,G is the matrix productBA.

Review Session

I will be happy to run a pre-exam review session next week. Is Wednesday, March 24 in the evening OK?

Review Problems

From Smith: Chapter 8/1,2,5,17,18,22; Chapter 9/13 and also compute the matrix of S with respect to the bases {1,x,...,xⁿ} in the domain and codomain; Chapter 10/6,11,12,14,18 (you will need to look up the definitions of these matrix concepts in the text; if I asked you something like this, I would provide the definition(s)),37; Chapter 11/1-5,7-12,21,23,24