Math U371 Linear Algebra
Class Notes, Fall 2006
Prof. A. Iarrobino
Meeting Mon, Tues, Thurs 9:15-10:30 in 456 Ryder.
Math U371 syllabus Syllabus A, with basic assignments.
Math U371 information (pdf) Syllabus B,
information about the course, grading, policies.
Math U371 info (Word)
Plan of classnotes web page:
Look here for specific assignments and brief comments on what we did (or will do) in class. Also for resources such as sample quizzes, exams,
and links to online resources. I plan to update this webpage by 5 PM after each class.
TBA = "to be announced". Note that I will give a minimal assignment, and further problems for practice,
you don't need to do all the "further" problems, but you are invited to try them. At times I will collect and mark the
homework (+,-, OK).
In html I will write matrices giving each row with entries separated by commas, then semicolon, then next row. The notation (1,2,3)^T
denotes the column vector with entries 1,2,3. The notation \in means "element of" (from TeX).
Week of September 6: Geometry of linear equations, Gauss-Jordan elimination.
First Meeting: Wednesday September 6 at 9:15 AM in 456 Ryder. (Plan). Go over information. Section 1.1 Linear systems and their geometry.
Homework for Thursday Sept 7 and Monday Sept 11. Section 1.1 p. 5 #1,7,10,20,21,34, 40.
Thursday September 6: (Plan). Questions: 1.1 #40, done intuitively (using that one point has z-coordinate zero), and by Gaussian reduction of
a system of 2 equations in four unknowns (the coefficients a,b,c of x,y,z, and the constant term d).
The system was a+b+c-d=0, and 3a+5b-d=0 Or (1,1,1,-1 | 0); (3,5,0,-1 | 0). Rref was (1,0,2.5,-2|0);(0,1,-1,5,1|0).
Taking arbitrary values of c,d, one may solve these for (a,b), obtaining the equation of a plane ax+by+cz=d containing the two points (1,1,1), (3,5,0).
Row reduction defined. rref (row-reduced echelon form) defined.
We prepared for Section 1.2 on GJ reduction.
Homework for Monday Sept 11:
A. Section 1.1. Complete above problems, also do Further problems Further: 26,27,44,46.
B. Read Section 1.2 p. 20, in preparation for Monday's class. Begin #5,7,18,29-31.On #5,7, do not worry too much about doing the GJ reduction, just find row
operations to get the system to rref, as we will study GJ on Monday. But if you can already do the GJ reduction after reading Section 1.2, then so much the better!
Week of September 11: Continue Gauss-Jordan row reduction, matrices. Quiz # 1, Thursday.
Monday September 11: Gauss-Jordan row-reduction to rref(A) (Section 1.2)
Setup of linear equations: Section 1.1 #20.
HW for Wednesday Section 1.2 #2,4,5,7,20-22,29-31,34,35,41 (from syllabus).
Wednesday September 13: Students put #2,4,5,7 on board, discussion of Gauss-Jordan rules (text p. 18),
versus general row reduction (eyeball and simplify) to rref.
Interpreting solution using parameters.
Rank of matrix (number of leading ones). Inconsistent systems, Geometry of solution space.
Scalar product of row and column vector, writing system as AX=C, X=variable vector, C=constant vector. Nullspace of A (solve system AX=0 vector).
Worksheet #1.
Thursday September14: (Prof. Gordana Todorov substituting this class only) Questions, Quiz #1.
Homework for Monday, September 18 Read Section 1.3. Section 1.3 #1-8, 10-15. Also, Worksheet #1 (correcting last constant
of vector b in #1 problem from 6 to 4, then solution is u=v=w=1.
Week of September 18 Solutions and matrices. Linear transformations and inverses.
Monday September 18: Went over Quiz 1. Solutions and matrices, matrix product (section 1.3).
System AX=C.
Relation between nullspace (solution to AX=0) and solution set to linear system AX=C. Geometrical interpretation of linear
system and its solution
Homework for Wednesday Sept 20: Section 1.3 #1-8, 10-14, WS #1.
Wednesday September 20 Matrix Algebra, continued.
solutions to AX=C and the relation with the nullspace,
Solutions to AX=0, continued, with examples. The nullspace is a linear space through the origin;
the solution set has the form X=X_0+N, a point, line, plane or k-dimensional fold through a point X_0,
parallel to the nullspace N. Parallelogram law.
Homework for Thursday September 21 Section 2.3 #21-32,34,36,47,55.
Thursday, September 21 Role of linear maps as approximations to multivariable functions. Rank of a matrix, parallelogram law, continued. Solving vector equations,c_1v_1+c_2v_2=v_3, for
constants c_1,c_2.
HW for Monday: Read Section 2.1, linear transformations. Try p. 50 #1-4.
Week of September 25
Linear transformations T from R^m to R^n, given by nxm matrix M.
Second definition of linear transform (or major properties). (Section 2.1)
Geometric properties of common linear transforms from R^2 to R^2. (Section 2.2)
Inverse
transform, when m=n= rank M_T. (Section 2.3).
Properties of matrix products( Section 2.4).
Intro to kernel and range of T (Section 3.1-3.2). Quiz #2 Thursday.
Monday September 25 Linear transformations, defined by a matrix, basic properties. Standard coordinates on R^m, columns of M_T as transforms
of the standard coordinates.
geometric transforms of R^2. (Section 2.1,2.2).
Note
No office hour Monday Sept 25.
Normal or extended office hours Wed Sept 27, Thurs Sept 28.
Homework for Wednesday, September 27. Section 2.1 p. 50 #1-6,9,24-30,35.43-45.50.
Wednesday September 27 Sections 2.1,2.2. Matrix products and linear transforms. (Section 2.4)
Homework for Thursday, September 28.
Section 2.2 p. 66 #1,4,6-10,17.19.21,23,25,26,49.
Section 2.4 p. 89. #3,5,11,13,16-25.
Quiz #2 from Winter 2003 Math 1301 (practice for Quiz 2)
MTH 1301 Winter 2003 Quiz 2 Solutions (pdf file)
Thursday September 28 Questions on geometry of linear maps, products and Sample Quiz 2.
Intro to kernel and image (nullspace and
column space, again!) Section 3.1
Homework for Monday, October 2
Section 2.3 p. 76, #1-5,17,19,35-41 (odd only).
Section 3.1 p. 109 #1,3,5,7,10,14,15,16,23,25,33,35,44,46.
Monday October 2 Questions on practice quiz 2 (15 min), quiz 2 (50 min). Quiz 2 take home
portion handed out, due Wednesday Oct 4.
HW for Wednesday Oct 4
Section 3.1 p. 110 #24,27,34,46,48, 50*, 51*.
Read Section 3.2 (at least start), Begin p.121 #1-3,6.
Also, take home portion of Quiz 2 (one page)
Extra Office Hour Tuesday Oct 3, 10:40-11, 12-12:30.
Wednesday October 4
Subspaces, span, linear independence, bases of a space. (Section 3.2).
We will also decide on the day for Exam 1.
HW for Thursday October 5 Section 3.2 p. 121 #11-33 odd, 24, 34.
Thursday, October 5 Dimension of a subspace of R^n (section 3.3).
HW for Wed October 11 Section 3.3 p. 133, #1,3,5,7,11,13,17,21,23,27-30,36-39.
Week of Oct 9 No class Monday Oct 9 (holiday). Chapter 3, begin Section 5.1.
Wednesday Oct 11 Check HW, form (optional) Class Work Groups, linear combinations, bases
(Section 3.3).
Homework Section 3.3 as above.
Note: We skip Section 3.4 and Chapter 4
Thursday Oct 12 Questions on Sections 3.2, 3.3 HW. Orthonormal Bases (Section 5.1).
HW for Monday Oct 16
Section 5.1 #1,3,5,13,15,17,27,35.
Also, the Sample Exam 1 (from Winter 2003) posted below.
Week of October 16 Prepare for Exam 1, Exam 1 Wednesday, Gram-Schmidt process (finding
orthonormal basis), QR factorization. Orthogonal Transformations (?).
Monday October 16 Questions on Sample Exam, sample problems
(if time) Gram-Schmidt process.
Wednesday October 18. Exam 1.
HW for Thursday October 19 (If discussed Monday) Section 5.2 #5,7,19,21,33,35.
Thursday October 19 Exam 1 back. Gram-Schmidt Process, QR Factorization. (Section 5.2).
Homework for Monday October 23
Section 5.2 p. 208 #5,7,19,21,25-28,32,33,35.
Week of October 23 Orthogonal Transformations, Least squares, Inner product spaces.
Monday October 23 Orthogonal Transformations (Chapter 5.3). Five equivalent characterizations
of A (nxn matrix) defining an orthogonal transformation. Properties of OT: A,B orthog nXn
imply AB, A^T=A^{-1) also orthogonal, but not A+B.
Homework for Wednesday
Section 5.3 p. 216 #5-8,13-17,27-29. 33,34
Wednesday October 25 Questions on OT nxn.
New: mxn OT, matrix of orthogonal projection
(Warning OP is not OT), section 5.3
Least Squares and the equation A^T A X= A^T b. (section 5.4).
Now a word from our sponsor for applications oriented readers: Least Squares is a really neat application of what we have learned in linear algebra,
with many uses. See the problems below in Section 5.4 on LS
Homework for Thursday
Section 5.3. #29-32, Extra 39-44.
Section 5.4. p. 228 #1,2,5,7,8,11,13,17-25, 31-33, Extra 38-39.
Thursday October 26 Least Squares continued,
Week of October 30 (Retrospective) Determinants, expansion of determinant by rows, columsn,
Properties of determinants (Sections 6.1,6.2), eigenvalues (section 7.2). Brief Quiz Thursday on Least squares solutions.
Homework, week of October 30
Section 6.1 #1-11 odd, 17,27.
Section 6.2, 1,6,24-25,29,31,49,50* (optional).
Homework for Monday November 6 Section 7.2 #1-3.
Week of November 6 Eigenvalues, eigenvectors, eigenspaces. Discrete dynamical systems.
Brief quiz Thursday (determinants, eigenvalues, eigenvectors).
Monday November 6 Questions, eigenvalues; eigenvectors, eigenspaces. Geometric interpretation of eigenvectors.
Singular matrices (rank < n) have zero as eigenvalue.
Homework for Wednesday Nov. 8
Section 7.2 #4-13 odd. (eigenvalues)
Section 7.3 #1-13 odd (eigenvectors, eigenspaces.
Wednesday November 8: Questions on eigenspaces.
Eigenvalues for matrix powers and matrix polynomials;
application to discrete dynamical systems.
Thursday Nov. 9 Expansion of determinants (examples). Brief Quiz B.
Week of November 13 (Retrospective): Determinants (anon), eigenvalues (Section 7.2).
Diagonalization, similarity, characteristic polynomial invariant under similarity(Section 7.3).
Homework, week of Nov. 13:
Section 7.4 #1,3,5,17,31,33,35,36,41,60.
Section 7.1 # 1-7,9,15-22,34,51-54 (DDS).
Comments, Summary on diagonalization (from e-mail to class, Nov. 16)
A. Comments Nov. 15 were almost uniform that the new material was
challenging ("went past me", or "don't understand Jordan form" and wish for more on the new material, Jordan blocks, and on algebraic and geometric multiplicity.
B. So today (Thurs. Nov. 16) was on these topics, with examples.
a. Similarity, its definition, why similarity is
an equivalence relation. (See fact 3.4.6 p. 146, in Section 3.4,
we showed a,b,c in class.)
b. And the connection of similarity to characteristic polynomial
f_A(\lambda)= det(A-\lambda I) [book notation, I used
p_\lambda (A) in class ).
The characteristic polynomials of similar matrices are the same
(Fact 3.7.6 a. p. 323, see proof of (a).there)
c. Jordan form: if all roots of char polynom of A are in the
base field R (so char polynomial factors completely), then
i. A is similar to a matrix in Jordan normal form (JNF)
(Jordan blocks down the main diagonal).
Jordan blocks defined in class (see also downloadable file:
Jordan Normal form for 2x2, 3x3 (University of Warwick)
which is very well done, and can be skimmed for "Jordan blocks"
ii. B is similar to A iff B is similar to the same JNF.
So JNF classifies such matrices (those with all eigenvalues real) up to similarity.
B1. We used this to answer #35-37 p. 338 (all "yes", since
they are 2x2 and have the same characteristic polys with 2
distince eigenvalues.
B2. If A is similar to diagonal matrix D, then the power A^n can
be calculated from D^n, using the eigenvector matrix S
(see Example 4 p. 333 of text), and
"Algorithm 7.4.5" p. 334.
- This is useful for finding x(t) in DDS (see 7.1 Example at start,
and problems 7.1 #52. 53.)
C. Monday Brief quiz will be on recent topics (text and class):
a. eigenvalue, eigenvectors, characteristic polynomial
b. similar matrices, (definition, properties)
c. diagonalization vs Jordan normal form
d. powers of a matrix using diagonalization
e. DDS (discrete dynamical systems)
Monday Nov. 20 Questions on diagonalization.
Complex numbers, eigenvalues. (Section 7.5)
Brief Quiz C
Brief Quiz C (pdf)
Brief Quiz C, solutions (pdf)
HW for Monday Nov. 27. Read Section 7.5 (complex eigenvalues) p. 350 #1,2,5,6,8,20,23.
Read Section 7.6. p. 359 #1,2.
Wednesday Nov. 22 Questions, went over Brief Quiz, (optional class).
Week of Nov. 27. Questions/examples on diagonalization, Jordan normal form.
Questions on complex eigenvalues and stability (Sections 7.5,7.6).
Symmetric Matrices (section 8.1), Singular value decomposition (Section 8.3) (these are our last topics)
Thursday Nov. 30: Quiz 3. (see below for topics)
Homework for week of Nov. 27 (as assigned)
Section 7.2 p. 314 #24-27.
Section 7.5 (See above, HW for Nov. 27).
Section 7.6 # 1,2,11,17.
Section 8.1 (symmetric matrices) #1,3,7.
Monday November 27 Went over Brief Quiz C, and Jordan Normal form.
Complex eigenvalues, stability for 2x2 matrices (section 7.5, 7.6).
HW for Wednesday See Section 7.5, 7.6 HW above. Read Section 8.1.
Week of Dec. 4 (Monday, Wednesday). Singular value decomposition. Questions, review.
Monday Dec. 4 Singular value decomposition (Section 8.3).
HW for Wednesday Section 8.3 (SVD) #1,2,4,6,12-14.
Wednesday Dec. 6 Questions, review for final exam.
Forthcoming events, attractions
FINAL EXAM-Monday December 11: The regular final exam in Math U371 is rescheduled to Monday Dec. 11 at 10:30-12:30 AM. Room 43 SL
Alternative times: please arrange with me in advance.
For a sample final, download the
Spring 06 Math U371 Final (pdf file) from the bottom of the syllabus web page:
Math U371 syllabus (no solutions on web)
Prof. Suciu's web page for Spring 2005 Linear Algebra: Math U371 Linear Algebra: Prof. Suicu
2005 has Sample Final with downloadable solutions and also an actual final with downloadable solutions, (go to the bottom of the link).
Retrospective Events
Quiz 3 Thursday Nov. 30. Topics: See Brief Quiz A-C; also complex eigenvalues, stability,
and Spectral theorem (section 8.1).
Sections 6.1,6.2, Chapter 7 (main emphasis), plus JNF, Section
8.1.
Math 1301 W03 Quiz 3 (pdf)
Math 1301 W03 Quiz 3 Solutions (pdf)
Brief Quiz C Monday Nov. 20 (similar matrices, characteristic polynomial, algebraic and geometric multiplicity, diagonalization,
JNF (Jordan Normal form), DDS (Discrete Dynamical Systems)
Brief Quiz B Thursday Nov. 9 (determinants, eigenvalues, eigenvectors).
Brief Quiz A Thursday Nov. 2 (least squares solutions)
Partial make up for Midterm Exam (maximum B+): Week of October 30, Wed. or Thurs afternoon.
Directions to be passed out Wednesday Oct. 25.
Exam 1 Wednesday Oct. 18, Chaps 1-3, up to and featuring 3.3. Section 5.1.
Sample: Exam 1 from Math 1301, Winter 03 (Ignore #4D)
Solutions: Solutions to Exam 1 from Math 1301,
Winter 03
Quiz #2 Monday October 2. Systems of linear equations AX=C, their solutions, and their geometric and column space
interpretation. Linear transformations and their inverses (when they exist).
Geometric properties of Linear Transformation. Null space = kernel, column
space, rank, dimension of kernel. Product of matrices. Section 1.3, Chapter 2, and Section 3.1.
Take home portion due Wed Oct 4.
Quiz #1 Thursday September 14:
Topics: A. Obtaining a system of linear equations from word problem
B. Row reduction, Gauss-Jordan row reduction of system to rref. (includes recognizing rref!)
C. Interpreting/solving rref, using parameters, when appropriate
D. Geometry of system of linear equations. Dimension of solution set.
To prepare see sections 1.1, 1.2 of text, and classwork. Also see Worksheet #1.
Questions or comments: e-mail to a.iarrobino@neu.edu.
This is usually the quickest way to reach me, or come by 526 NI, MWTh afternoons
Prof. Anthony Iarrobino
Department of Mathematics
Northeastern University
Boston, MA,
02115
Office:526 NI
Phone: (617) 373-5524
Email: a.iarrobin@neu.edu
Created: September 4, 2006. Last modified: December 7, 2006 .
URL:http://www.math.neu.edu/~iarrobino/AIMathU371.F06.classnotes