AMa 104 (Matrix Theory)

Course Topics: Linear operators on normed and inner-product spaces, matrix theory
Instructors: Dr. M. Holst
Term: Fall 1996

Outline: Matrix Theory: Linear Operators on Finite-Dimensional Spaces

HOMEWORK #1 can be found here (10-8-96).

DUE DATE: Homework #1 is do at 2pm on 10-17-96.

HOMEWORK #2 can be found here (10-22-96).

DUE DATE: Homework #2 is do at 2pm on 10-31-96.

NOTE: (10-27-96) There was an error in the statement of Problem 7 in the original HW2; I left out various characters in the definition of the little L2 norm. This mistake is corrected in the verions of HW2 posted here.

NOTE: (10-28-96) Problem 6 of HW2: since the Fredholm alternative doesn't address uniqueness in the entire space, don't worry about the uniqueness part (or construct two solutions, showing that you don't have uniqueness).

NOTE: (10-28-96) Problem 4 of HW2: I'm just asking you to construct the matrix representation of the linear transformation which maps the first two vectors (v1, v2) to the second two (w1,w2). This problem is nearly identical to Example 4 on page 119 of Strang's book. You simply need to determine what the two columns of the [3x2] matrix A need to be, so that the linear combination of the columns specified by v1 produces w1, and that the linear combination of the columns specified by v2 produces w2. You have exactly enough conditions to determine all of the entries of the [3x2] matrix A.

For the second part, when the vectors (v1,v2) change and (w1,w2) stay the same, then the mapping from (v1,v2) to (w1,w2) is clearly a different linear transformation than for the original choice of (v1,v2). All I want is for you to construct its matrix representation, in exactly th same way.

Working with two bases like this is required when the linear transformation maps one space to a different space; when the spaces are the same, it suffices to pick a single bases with which to express the linear transformation as a (then square) matrix. When working with such a linear transformation on the same space, it is common to use the canonical basis to represent the matrix.

Note that if we had taken (v1,v2)=(e1,e2), the canonical basis in R2, and then taken (w1,w2)=(e1,e2), the first two vectors of the canonical basis in R3, then the matrix representation of the transformation taking (v1,v2) into (w1,w2), respectively, would simply be the matrix with columns a1=(1,0,0) and a2=(0,1,0).

HOMEWORK #3 (midterm take-home exam) can be found here (11-1-96).

DUE DATE: Homework #3 is due at 12pm (noon) on 11-8-96.

GRADING: Homework #3 is worth the same amount of points as a typical homework.

NOTE: (11-1-96) This exam/homework is similar to the previous two homeworks in difficulty (it is a bit shorter since you don't have quite as much time to do it as I usually give you). I'm just calling it an "exam" because I want you to write up your results individually on this homework (again, you are encouraged to work together on all of the other homeworks).

NOTE: (11-3-96) A very on-the-ball student (well, that could be any one of you I guess) pointed out the severe problems with the extra-credit problem (Problem 3); I made a total mess of that problem by accidentally combining what should have been two parts into one. The fixed homework is posted now...

HOMEWORK #4 can be found here (11-19-96).

DUE DATE: Homework #4 is due at 12pm (noon) on 12-5-96.

NOTE: (11-20-96) Forgot to link the page to the homework; you should be able to reach it now...

HOMEWORK #5 (final take-home exam) can be found here (12-5-96).

DUE DATE: Homework #5 is due at 12pm (noon) on 12-13-96.

GRADING: Homework #5 is worth the same amount of points as a typical homework.

NOTE: (12-5-96) This exam/homework is similar to the previous four homeworks in difficulty. As I did for Homework #3, I'm calling it an "exam" because I want you to write up your results individually on this homework, although you are welcome to discuss the problems together.

NOTE: (12-5-96) A couple of crises have delayed my getting this typed in; I'll have it posted by tomrrow evening (12-6-96). (I'll shorten it a bit since you have one day less to complete it.)

NOTE: (12-8-96) Typo: On the first line of Problem 2, J_p should be J_k.

NOTE: (12-9-96) Typo: On the third line of Problem 1, M and it's inverse should be reversed; you all know from class that these two expressions should be equivalent to: AM=MJ. The rest of the problem uses this correct expression.