Linear Transformations and Matrices (Math 315) for Spring 1997

Spring 1997

Linear Transformations and Matrices (Math 315, E2)

260 Mech. Eng. Bldg, 1pm, MWF

Instructor: Mihail Kolountzakis


Prerequisites: MATH 242 OR MATH 245.

Text: S.J. Leon, Linear Algebra with Applications, 4th Ed., Prentice Hall, 1994.

Material to be covered (approximately): Chapters 1-6. (Not everything will be covered from those chapters.)

Topics include: Methods of solving linear systems, determinants, matrix algebra, vector spaces (linear independence, bases, dimension), linear transformations, orthogonality (least squares, the Gram-Schmidt orthogonalization process), eigenvalues and diagonalization.

Grading policy: 25% each of two midterm exams, 50% for the final. Homework might help push up in marginal cases.

Office hours (at 222 Illini Hall): M, 3-4pm or by appointment.

You may want to have a look at last semester's three tests. The files are in Postscript. On a UNIX system you can see them with "ghostview" and send them directly to the printer.
(Here is some information on how to view Postscript files on a PC or Mac system.)
1st midterm, 2nd midterm, and final.

W, Jan 22: § 1.1, 1.2 (intro. to linear systems, elementary row operations that do not change the solutions of a system, a few examples of how to use row operations to solve a linear system).

F, Jan 24: § 1.2 (how to bring an augmented matrix of a linear system to row exchelon form , examples of finding the solutions of a linear system which is in row echelon form).

Homework Grading Policy: Turn in all the assigned problems on the date they are due. Some of them will be graded by either myself or the TA or both. We may write solutions for some of the problems. Most of the assigned problems will end up being discussed in class.
--Remember that there is no direct influence of the homework grades on your final grade, but doing it is the only way to prepare for the tests.
--Try to be brief and precise. Don't just list formulas and numbers but throw in some English as well to make it readable (by the grader and by yourself later on). Good style will be rewarded.

Homework 1: Chapter 1
§ 2: 1,2,3,6,12,13
§ 3: 1,2,6,9,10,12,13
§ 4: 7,8,11,17,18,20
Due in class on M, Feb 3.

Teaching Assistant: Jonathan Hayward
Office Hours: 2-3 MW at 48 Noyes.

M, Jan 27: § 1.3: introduction to matrices, operations with them (addition, multiplication by a scalar, multiplication of matrices, vectors, transpose of matrix), algebraic rules for matrix operations.

W, Jan 29: § 1.4: Diagonal and triangular matrices, inverse matrix, matrix form of a linear system, elementary matrices corresponding to the three kinds of row operations, non-singular systems.

F, Jan 31: § 1.4: Continuation of Theorem 1.4.3 and the Corollary. Examples and solutions of several problems from § 1.4.

M, Feb 3: Homework due today in class. Solutions were distributed (leftovers are outside my office). We covered § 2.1 (determinants).

Homework 2: Chapter 2
§ 1: 3,4,5,11,13
§ 2: 3,5,6,7,10,12(do this one also without using determinants),16
§ 3: 2,4,7,8,10,11,12,13.
Due in class on W, Feb 12.

W, Feb 5: § 2.2, 2.3: properties of determinants, their use in solving linear systems.

TUTORING AVAILABLE: Free tutoring is available for students of Math 315 (Prof. Irma Reiner).
Mon, Tue, 2:30-4pm, Thu 2:30-3:30pm at 315 Altgeld Hall.
There might also be some tutoring available in your Residence Hall. The schedule is posted outside my office.

F, Feb 7: Problems from Chapter 2, and question answering. Come prepared to ask questions about what you don't understand.

M, Feb 10: § 3.1. Vector spaces.

Homework 3: Chapter 3.
§ 3.1: 1, 4, 5, 10, 11, 15, 16
§ 3.2: 1, 2, 3, 4(a,b), 5, 6, 8, 10, 11, 14, 15, 19
Due M, Feb 17, in class.

1st Test tentatively scheduled for M, 24 Feb.
Room and time TBA.
Material to be examined: Chapters 1,2 & 3.

W, Feb 12: § 3.2. Subspaces of vector spaces. The linear span of a collection of vectors. The Nullspace N(A) of a matrix A.

F, Feb 14: § 3.3: Linear independence of vectors. Relation of linear independence to the singularity of a matrix.

Homework 4: Chapter 3.
§ 3.3: 2(a,b,c), 4(b), 5(c), 6, 10, 13, 16
§ 3.4: 3,4,8,10,13,14,16
Due M, Feb 24, in class.

2nd Homework graded: I remind that you do not get any direct grade from your homework sets (except in marginal cases and that's why you should hold on to the graded hwk papers until the end of teh semester). However I am trying to grade a few problems and make some remarks on your papers that are supposed to help you understand where you stand. So, please, do turn in some hwk (and also come to pick it up).
Specific comments: I graded problems 1.5, 1.11, 2.7 and 3.8 (Chapter 2). Almost everybody got problems 1.5 and 2.7 right. A great many people have had problems with 1.11 and, to a lesser extent, with 3.8. The typical mistake was that people would try to prove parts 1.11(b) and (c) by giving an example. However an example can only disprove a statement, not prove one. So, in 1.11(a) where you want to show that det(A+B) is not equal to det(A)+det(B), for all 2x2 matrices A and B, it is enough to produce an example of two such matrices for which the equality does not hold. So, in this case you're disproving an equality by giving an example. But you cannot do the same for 1.11(b) or (c) because there you want to show that the equality involved is actually true. In those cases you need to give a general argument that works for every two matrices (refer to the solutions).

M, Feb 17: § 3.4: Basis and dimension of a vector space.

1st Test: M, 24 Feb., in 223 Greg Hall from 7pm-8pm.
Material to be examined: Chapters 1(except § 1.5),2 & 3(up to and including § 3.4).

W, Feb 19: § 3.4: Continuation.

F, Feb 21:§ 3.5: Change of basis. Solutions for the homework that was due Monday will be distributed.

M, Feb 24: Review. Come armed with questions you want answered. Do not forget the TEST tonight.

W, Feb 26:§ 3.6: Row space and column space.

F, Feb 28:Graded test papers will be distributed in class. The problems will be discussed. We continue the discussion about the row and column spaces of a matrix.

Homework No 5:
§ 3.5: 1,2,5,6,9
§ 3.6: 1,3,6,7,11,14,15,17,18,19,20
§ 4.1: 1,4,5,6,7,11,15,16,19,22
§ 4.2: 1,2,3,4,5,8,13,14
Due W, March 12, in class.

M, Mar 3: § 4.1: Introduction to Linear Transformations.

The Average score for the first test was: 32.35.

W, Mar 5: § 4.2: Matrix Representation of Linear Transformations.

F, Mar 7: NO CLASS.

M, Mar 10: Problems and examples from § 4.2. If time permits introduction to § 4.3.

W, Mar 12: Similarity of matrices (§ 4.3).

F, March 14: § 5.1. The scalar product (dot product, inner product) in Rn.

On F, March 14, class be held at 161 NOYES LAB.

Homework No 6:
§ 4.3: 2,4,5,10,11,12,14,15
§ 5.1: 6,7(a,b),8,10
§ 5.2: 1(d),2,3,5,7,9,13,15
Due F, Apr 4, in class.

Comments on graded homeworks No 4 and 5:
Many people did not understand some of the questions (I looked at § 3.6.14 and § 4.2.13). I don't remember any of them asking me about them in class or during office hours.

M, March 17: § 5.2: Orthogonal subspaces, orthogonal complements.

No Class will be held on W, March 19, and F, March 21. We resume normally on M, March 31.

M, March 31: Prof. Harold Diamond will teach the rest of § 5.2.

The Final Exam will be held in 165 Everitt, on the 16th of May, from 1:30-4:30pm.

W, April 2 and F April 4: § 5.3: Vector spaces with inner products and norms.

M, April 7 and W, April 9: Problems.

Homework No 7: Chapter 5
§ 5.3: 2,4,7,8,9,13,14,17,19,20,21,26,28
§ 5.4: 1(a),2,3(a),5,6,7,10
§ 5.5: 1,4,5,6,7,9,10,12,13,14,17(a,b(ii)),18,21,23,27,29.
Due in class, M April 21.

F, April 11: § 5.4: Least squares.

M, April 14 and W, April 16: § 5.5: Orthogonal sets and matrices.

Second Test: Will be given in class on F April 18. Study up to and including § 5.3.

The average score for the second test was 35.6.

M, April 21, and W, April 23: § 5.6: The Gram-Schmidt orthogonalization process, QR factorization (NOT the modified G-S process).

Homework No 8: Chapter 5
§ 5.6: 5, 6, 7, 12.
Due in class on W, April 30.

F, April 25, and M, April 28: § 6.1: Eigenvalues.

W, April 28: Problems from § 6.1.

F, May 2, and M, May 5: § 6.3: Diagonalization.

Final Homework: This will not be collected but do the problems as practice for the final exam.
§ 6.1: 1(a,b,g),2,5,6,7,8,9,10,11,13,15,19,24.
§ 6.3: 1(a,d),5,6,7,8(c,d),9,11,12,13.

W, May 7: Review.

The Final Exam will be held in 165 Everitt, on the 16th of May, from 1:30-4:30pm.

The first test, the second test, and the final.

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