Course Description:

Systems of linear equations, matrices, finite dimensional vector spaces, determinants, eigenvalues and eigenvectors.

Instructor:

Aaron Nung Kwan Yip

Department of Mathematics

Purdue University

Contact Information:

Office: MATH 432

Email and Phone: click here

Lecture Times and Places:

Section 042 (CRN 19439): T, Th 12:00pm - 1:15pm, SCHM 316

Office Hours:

T, W: 4:30pm - 6pm, MATH 432, or by appointment.

Occasionally, due to unexpected events, there is a need for online meetings and lectures. These will be conducted in Zoom.
You can also find this link in Brightspace MA351 course homepage (upper left corner, second tab): Content/Course Materials.
This link will also be used in case you need to see me online.

Textbook:

Main Text (required):
[P] Linear Algebra, Ideas and Applications, 4th edition, Richard Penney, Wiley.
You are highly encouraged to make good use of the textbook by reading it.

Homework:

Homeworks will be assigned weekly, due usually on Thursday in class. They will be gradually posted as the course progresses. Please refer to the course announcement below.

Steps must be shown to explain your answers. No credit will be given for just writing down the answers, even if it is correct.

As a rule of thumb, you should only use those methods that have been covered in class. If you use some other methods for the sake of convenience, at our discretion, we might not give you any credit. You have the right to contest. In that event, you might be asked to explain your answer using only what has been covered in class up to the point of time of the homeworks or exams.

As a rule of thumb, you should make use of all the information given in a problem. No point will be given by just writing down some generic statements, even though they are true.

Please staple all loose sheets of your homework to prevent 5% penalty.

Please resolve any error in the grading within one week after the return of each graded assignment.

No late homework will be accepted (in principle).

You are encouraged to discuss the homework problems with your classmates but all your handed-in homeworks must be your own work.

Examinations:

Tests: Midterm One (Week 6, Sept 26th), Midterm Two (Week 12, Nov 7th), both in class

Final Exam: During Final Exam Week

No books, notes or electronic devices are allowed (nor needed) in any of the tests and exam.

Grading Policy:

Homeworks (25%)

Test (40%, 20% each test)

Final Exam (30%)

Class Participation (daily or weekly quizzes, etc, 5%)

You are encouraged to attend all the lectures. However, I do not take attendance. The quizzes are used to check your basic understanding and provide opportunity for you to mingle with your classmates and myself. It is open book, open note and open discussion, hopefully a fun activity. No make-up quiz will be given. You do not need to worry if you miss a few. However, if you anticipate to miss more (for legitimate reasons), please by all means let me know as soon as possible.

The following is departmental policy for the grade cut-offs:
97% of the total points in this course are guaranteed an A+,
93% an A,
90% an A-,
87% a B+,
83% a B
80% a B-,
77% a C+,
73% a C,
70% a C-,
67% a D+,
63% a D, and
60% a D-.
For each of these grades, it's possible that at the end of the semester a lower percentage will be enough to achieve that grade.

You are expected to observe academic honesty to the highest standard. Any form of cheating will automatically lead to an F grade, plus any other disciplinary action, deemed appropriate.

Nondiscrimination Statement:

This class, as part of Purdue University's educational endeavor, is committed to maintaining a community which recognizes and values the inherent worth and dignity of every person; fosters tolerance, sensitivity, understanding, and mutual respect among its members; and encourages each individual to strive to reach his or her own potential.

Student Rights:

Any student who has substantial reason to believe that another person is threatening the safety of others by not complying with Protect Purdue protocols is encouraged to report the behavior to and discuss the next steps with their instructor. Students also have the option of reporting the behavior to the Office of the Student Rights and Responsibilities. See also Purdue University Bill of Student Rights and the Violent Behavior Policy under University Resources in Brightspace.

Accommodations for Students with Disabilities and Academic Adjustment:

Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are also encouraged to contact the Disability Resource Center (DRC) at: drc@purdue.edu or by phone at 765-494-1247.

If you have been certified by the DRC as eligible for accommodations, you should contact me to discuss your accommodations as soon as possible. See also Courses: ADA Information for further information from the Department of Mathematics.

Campus Emergency:

In the event of a major campus emergency or circumstances beyond the instructor's control, course requirements, deadlines and grading percentages are subject to change. Check your email and this course web page for such information.

See also Emergency Preparedness and Planning for campus wide updates.

More information on University Policies:

See your MA351 course homepage in Brightspace.
Content (tab at upper left corner): Student Support and Resources, and University Policies and Statements.

Course Outline (tentative):

Chapter 1: linear systems and their solutions, matrices;

Chapter 2: vector spaces and subspaces, linear (in)dependence, dimension;

Chapter 3: linear transformation;

Chapter 4: determinants;

Chapter 5: eigenvectors and eigenvalues.

Course Progress and Announcement:

You should consult this section regularly, for homework assignments, additional materials and announcements.
You can also access this page through BrightSpace.

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Key outcomes of this course.
(1) setting up of systems of linear algebraic equations, finding their solutions, interpretation of solutions;
(2) effective use of matrix notations and their interpretation;
(3) interpretation of (1) and (2) using the concept of abstract (and yet concrete and useful) vector spaces, in particular, basis, dimension, and geometry of subspaces;
(4) last but not least, an introduction and initiation to the understanding and appreciation of the need of giving proofs, how to write proofs and knowing what constitutes a proof.

NOTATION MATTERS!!!!!!!!!!!!!!!
A clear understanding of notations is one of the keys to fullly appreciate mathematics.
The notations created for and used in linear algebra are supposed to make the concepts and computation easier.
But you need to UNDERSTAND them in order to get the most out of them.

READ THE TEXTBOOK!
Get used to how mathematics are formulated and presented.

My MOTTO on the use of technology (which I use often):
IF TECHNOLOGY HELPS YOU UNDERSTRAND, BY ALL MEANS USE IT. OTHERWISE, USE IT AT YOUR OWN RISK!
For the homework, I believe all the problems should be and can be done by hand. In order to get full credit, sufficient steps must be shown. You are welcome to use technology to check your answers.

BEWARE THAT DURING THE TESTS AND EXAM, NO TECHNOLOGY WILL BE ALLOWED.

Some matlab information.
(1) Matlab and linear algebra go hand in hand. Its effective usage
(a) requires good understanding of linear algebra, and also
(b) enhances your understanding of linear algebra.
(2) A very simple tutorial. Just follow the steps in the file.
(3) There are "lots" of Matlab manual available online. Type "matlab manual" in google.

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Week 1 (Aug 20, 22):

[P 1.2, 1.3]
Geometric interpretations of finding solutions:
(i) (row) intersection between lines, planes;
(ii) (column) writing vector as linear combination;
(iii) (map) finding pre-image of a point under linear transformation.
Elementary row operations (ERO):
(i) interchange two rows;
(ii) multiply a row by a nonzero number;
(iii) add a multiple of a row to another.

Note: Three interpretations of solving linear systems
Note: Gaussian Elimination
Note: Examples of solving mxn systems

Homework 1, due: Thursday, Aug 29th, in class.

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Week 2 (Aug 27, 29):

[P 1.3]
General mxn linear system: m equations in n unknowns. (Note: m might not equal n.)
Key concepts of Gaussian eliminations:

elementary row operations (ERO),

equivalence between systems (under ERO),

row echelon form (REF),

backward substitution,

pivot vs free variables,

reduced row echolon form (RREF).
Three possibililies upon solving mxn linear systems:
(i) unique solution (only pivot variables, i.e. no free variables);
(ii) infinitely many solutions (some free variables);
(iii) no solution (inconsistent)
Some applications of solving mxn linear systems:

interpolating polynomials;

traffic flows;

Leontief economic model

Homework 2, due: Thursday, Sept 5th, in class.

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Week 3 (Sept 3, 5):

[P 1.1] Vectors in R^n:
Vector addition, scalar multiplication;
Linear combinations and span and their interpretation in terms of solving linear system.
(Note that Span can be used as a noun or a verb, with different meanings.)
[P 1.4] (NOTATION MATTERS)
Matrix multiplied by a column vector and its linearity property:

A(X+Y) = AX + AY; A(aX) = aAX ==> A(aX+bY) = aAX + bAY;

(A+B)X = AX + BX; (aA)X = aAX ==> (aA + bB)X = aAX + bBX;
Linear system of equation in matrix form: AX=b;
Column space of A: Col(A) = Span{columns of A}
Null space of A: Null(A)={X: AX=0}
Homogeneous (AX=0) vs inhomogeneous (AX=b) systems;
Structure of solutions for AX=b (assume it is consistent):

X = p + Null(A), where p is a translation vector, or a particular solution.

Note: Column and Null spaces of a matrix

Homework 3, due: Thursday, Sept 12th, in class.

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Week 4 (Sept 10, 12):

[P 1.4]
AX=b is solvable if and only if b is in Col(A);
The solution of AX=b is unique if and only if Null(A)={0}, i.e. no free variables;
More Unknown Theorem: there are some free variables; for consistent system, there are infinitely many solutions;
More Equation Theorem: there are some b such that AX=b has no solution.
[P 1.1] General vectors space:
Vector addition, scalar multiplication, and their properties;
Common examples: vectors in R^n, polynomials, matrices, functions.
How to check if two spans or solution representations are the same.
[P 2.1] Linear dependence between a list of vectors:
one (or some) vector can be written as linear combination of the rest.

Note: Concept of a general Vector Space

Homework 4, due: Thursday, Sept 19th, in class.

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Week 5 (Sept 17, 19):

[P 2.1] linear dependence and independence.
Dependence relation/equation,
Consequence of linear dependence.
Redundant vectors, how to throw away all the redundant vectors.

Note: Linear Dependence and Independence

Practice problems for [P 2.1] (no need to hand in):
Section 2.1, p.108, EXERCISES:
2.1, 2.3, 2.6, 2.7, 2.11, 2.13, 2.15, 2.16, 2.17, 2.18.

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Week 6 (Sept 24, 26):

Midterm One: Thursday, Sept 26th, in class.
No calculator or any electronic devices are allowed (or needed).
Materials covered: Penney, Chapter 1 to Chapter 2.1.

The best way to review this (or any) exam is to:
(i) go over lecture materials,
(ii) read the textbook, and
(iii) go over the homework and quiz problems.

Quiz 1-5 Solutions
Hw 1-4 Selected Solutions
Fall 2021 Exam One
Spring 2019 Exam One

Midterm One Distribution
Midterm One Solution

Homework 5, due: Thursday, Oct 3rd, in class.
Section 1.4, p.89, EXERCISES:
1.114(b,c,d,e), 1.115, 1.116, 1.117, 1.118, 1.119, 1.131, 1.132, 1.133
(The textbook has provided an example of solution 1.114(a).)

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Week 7 (Oct 1, 3):

[P] Chapter 1.4
Subspace: closed under vector addition and scalar multiplication.
Examples of subspaces: Span, Col(A), Null(A).
"More Unknowns Theorem": A^(mxn): m < n: columns of A must be linearly dependent;
"More Equations Theorem": A^(mxn): m > n: there must be a vector b such that AX=b is not solvable.

Homework 6, 7:
Homework 6, due: Tuesday, Oct 15th, in class.
Homework 7, due: Thursday, Oct 17th, in class.

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Week 8 (Oct 8, 10):
(Oct 8: October Break)

[P 2.1, 2.2] Basis and dimensions.
How to find a basis? Two categories of methods: (i) Col(A), and (ii) Null(A).
For any vector/sub-spaces the dimension is unique while you can have different basis;
Dimension is the maximum number of lin ind vectors
Dimension is the minimum number of vectors that can span
Dimension is the effective number of degree of freedom
In an n-dim space, any n lin ind vectors must span,
In an n-dim space, any n vectors that span must be lin ind,
In an n-dim space, for any n vectors: linear independence <=> span <=> basis.

Note: Basis and Dimension

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Week 9 (Oct 15, 17):

[P 2.3] Col, Null, and Row spaces associated with a matrix.
dim(Col) = number of pivots = rank;
dim(Null) = number of free var = nullity;
Rank+Nullity = Total number of variables (Rank-Nullity Theorem)
relationship between rank and nullity with lineary independence;
relationship between rank and nullity with solvability and uniqueness of solution
Non-singular matrices, equivalent properties of non-singular matrices.
[P 3.1, 3.2]
Linear transformations and their matrix representations
Composition of linear transformations and matrix multiplication: [TS]=[T][S]
Beware of dimension compatibility: C^(mxn) = A^(mxl)*B^(lxn)
In general, AB is not equal to BA

Note: Col, Null and Row spaces of A
Note: Matrix multiplications and properties

Homework 8, due: Thursday, Oct 24th, in class.
p.145 EXERCISES: 2.76, 2.77, 2.78
p.157 EXERCISES: 3.1, 3.2, 3.5, 3.6, 3.7, 3.10, 3.11, 3.15
p.173 EXERCISES: 3.26, 3.27, 3.44
(For 3.44, find a basis for space of matrices B such that AB=BA.)

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Week 10 (Oct 22, 24):

[P 3.1, 3.2]
Properties of matrix multiplications: distributive, associative.
Diagonal matrices, identity matrices.
Geometric examples of linear transformations: projection, reflection, rotation;
[P 3.3]
Inverse of a matrix and how to find it.
Properties of inverse.

Homework 9, due: Thursday, Oct 31st, in class.
p.175 EXERCISES: 3.41, 3.42, 3.52
p.190 EXERCISES: 3.64(a--f), 3.72, 3.74, 3.75, 3.76, 3.77, 3.78

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Week 11 (Oct 29, 31):

[P Chapter 4] Determinant of a square matrix
Computation of determinant using co-factor expansion;
Computation of determinant using row reduction.

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Week 12 (Nov 5, 7):
Midterm Two: Nov 7th, in class
No calculator or any electronic devices are allowed (or needed).

Materials covered: Penney, Chapters 2 and 3, up to Homework 9
The best way to review this (or any) exam is to:
(i) go over lecture materials,
(ii) read the textbook, and
(iii) go over the homework and quiz problems.

Hw 5-9 Selected Solutions
Quiz 6-8 Solutions
Fall 2021 Exam Two
Spring 2019 Exam Two

Midterm Two Solution
Midterm Two Distribution

Homework 10, due: Thursday, Nov 14th, in class.
p.249 EXERCISES: 4.1(acdei) (use co-factor expansions), 4.4, 4.5
p.258 EXERCISES: 4.12(ace) (use row operations), 4.13 (ace, use column operations), 4.15, 4.16, 4.24, 4.25, 4.26

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Week 13 (Nov 12, 14):

[P Chapter 4]
Further Properties of determinants
Determinant as a multilinear alternating function of the rows (and columns) of a matrix.
Applications of determinants:

Area of parallelogram and volume of parallelepiped

Solution of AX=B (for invertible, square matrix A) (Cramer's Rule)

Formula for inverse of a matrix.

[P 5.1] Eigenvalues and eigenvectors
AX=lambda X: X must be a non-zero vector (though lambda can be zero).
Finding lambda and X.

Homework 11: due: Thursday, Nov. 21st, in class.

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Week 14 (Nov 19, 21):

[P 5.1, 5.2] Eigenvalue and eigenvectors:
AX=lambda X: X must be an non-zero vector (though lambda can be zero).
Examples with distinct and repeated eigenvalues.
Algebraic multiplities (m_i) vs geometric multiplicities (g_i): 1 <= g_i <= m_i

Deficient/defective eigenvalues and matrices: g_i < m_i

Non-deficient/non-defective eigenvalues and matrices: g_i = m_i
Linear independence of eigenvectors
Diagonalizable matrices and diagonalization of matrices.

Note: Eigenvalues and Eigenvectors

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Week 15 (Nov 26, 28):
(Nov 28: Thanksgiving Break)

[P 5.1, 5.2]
Application of eigenvalues/eigenvectors/diagonalization:
Use eigenvectors as basis.
Computing matrix powers.
Application to Markov Chain: equilibrium/steady state, AX=X

Note: Diagonalization and Applications

(More) Practice Problems for Chapter 5

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Week 16 (Dec 3, 5):

Summary of 351

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Week 17 Final Exam Week
Final Exam: Monday, Dec 9th, 8:00am-10:00am, BRWN 1154

Materials covered: accumulative, i.e. everything covered in lectures, homeworks, quizzes.
(You can use the above course log as a rough review sheet.)
The best way to review is to read the textbook, go over and understand the homework and quiz problems.
NOTATION MATTERS!!!
No calculator or any electronic devices are allowed (or needed).

Extra Office Hour: Saturday, Sunday, 4pm-5pm.

Selected Solutions for Hw 10, 11 and Chapter 5 Practice Problems
Quiz 9, 10 Solutions
Spring 2019 Final
Fall 2021 Final

Final Exam Solution
Final Exam Distribution