MCS-221 Problems: Fall 2002

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Last updated 12/11/02 9:00 p.m.

MCS-221 Problem Assignments: Fall 2002

Section	Practice problems*	Problems to turn in	Due date
6.1	1, 5, 7, 10, 11	2, 8, 16	T 12/3
6.1.1	1	6	F 12/6
6.2	3, 5, 11, 13	6, 14 (Cayley-Hamilton theorem), diagonalization diagram handout	F 12/6
6.3	1, 3	2, 4, 6 *Extra credit: 8, 17, 18, 19	T 12/10
6.4	1, 2, 3
6.5	2, 3, 9

* These are merely some suggested problems. You should work as many problems as necessary to master the ideas.

Unless otherwise indicated, the problems are from our text, Linear Algebra: Ideas and Applications, by Richard Penney.

Acknowledge your sources (people and texts).
In nontrivial problems, show how you get your answers.
Turn in neat, well-written solutions, not messy first drafts. Trim "fringes." Staple.
Do not copy collaborative solutions; write up solutions in your own words.
Turn in homework on time. Each class day late reduces the possible points by 25%.
Do extra credit problems entirely on your own.

Problem 1.1 #9
Problem 1.2 #22
Problem 2.2 #10
Problem 3.3 #18
Let V and W be vector spaces. Let T: V -> W be a linear transformation. Let U be a subspace of V. Prove: (a) T(U) is a subspace of W; (b) T^-1({0}) is a subspace of V. Do not assume that T is a matrix transformation.
Problem 4.2 #12 (The reference should be to Exercise 10.) Assume that the set S is nonempty.
Problem 4.2 #13
Problem 6.2 #14 (pre-Cayley-Hamilton Theorem)

Give a clear statement of the proposition to be proved.
Proofs should be written in complete sentences. Mathematical expressions should be embedded in a grammatically sensible way. Look at your math texts to see how this is done.
Every step in a proof is or can be justified by a reason. Valid reasons include assumptions, definitions, and previously established results, as well as principles of logic. The extent to which reasons are omitted and proofs abbreviated depends on the audience.
Clearly define for the reader all variables and terms that you introduce. Make the "quantification"--whether "for all ..." or "there exists--of variables clear.