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 011 (CRN 23346): T, Th 10:30am - 11:45am, LILY G401

Office Hours:

M, W: 2:45pm-4:00pm, MATH 432, or by appointment

Occasionally, due to unexpected events, there might be a need for online meetings and lectures. These will be conducted in Zoom.
You can also find this link in Brightspace MA351 course homepage Content/Zoom (upper left corner, second tab).
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.
(available online using your Purdue Career account)
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. Submitting identical work constitutes one form of cheating.

Examinations:

Tests: Midterm One (Week 6, Oct 2), Midterm Two (Week 12, Nov 13), 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:

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

Homeworks (25%)

Test (40%, 20% each test)

Final Exam (30%)

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 an 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.

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 26, 28):

[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.
Vector Algebra:
(i) vector addition;
(ii) scalar multiplication.
properties of vector operations.
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
Ref: Vector Algebra (Johnston, Intro. Linear and Matrix Algebra)
Note: Gaussian Elimination

Homework 1, due: Thursday, Sept 4th, in class.

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Week 2 (Sept 2, 4):

[P 1.3] General mxn linear system: m equations in n unknowns. (Note: m might not equal n.)
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 and only 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 linear system:

finding interpolating polynomials

an example from "Nine Chapters" (original version)

traffic flows [P, p.72]

Leontief input-output economic model

Note: Examples of solving mxn linear systems

Homework 2, due: Thursday, Sept 11th, in class.

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Week 3 (Sept 9, 11):

[P 1.1] Vector algebra in R^n:
Vector addition, scalar multiplication, and properties [P, p.12, Theorem 1.1];
Linear combination 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;
Column space of A: Col(A) = Span{columns of A}
Null space of A: Null(A)={X: AX=0}
Linear system of equations in matrix form, AX=b:

Solving AX=b is equivalent to express b as a linear combination of the columns of A;

AX=b is solvable if and only if b is in Col(A).

Note: Column and null spaces of a matrix

Homework 3, due: Thursday, Sept 18th, in class.

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Week 4 (Sept 16, 18):

[P 1.4]
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.
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 must be some free variables; for consistent system, there must be infinitely many solutions;
More Equation Theorem: there must be 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.

Note: General vector spaces

Homework 4 (revised), due: Friday, noon, Sept 26th, in MATH 432, slide under door.

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Week 5 (Sept 23, 25):

[P 2.1] linear dependence and independence.
Definition 1 vs Definition 2 (see Note below)
Dependence relation/equation,
Consequence of linear dependence (1):

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, 2.24

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Week 6 (Sept 30, Oct 2):
Midterm One: in class, Thursday, Oct 2

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.
No calculator or any electronic devices are allowed (or needed).

Quiz 1-4 Solutions
Hw 1-4, Practice Problems (2.1) Selected Solutions
Fall 2024 Exam One
Fall 2021 Exam One

Midterm One Distribution
Midterm One Solution

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Week 7 (Oct 7, Oct 9):

[P 2.1] linear dependence and independence.
Consequence of linear dependence (2):

non-uniqueness of representation in linear combination.
[P 2.1] basis

linear independence and spans the whole space.
[P 2.2] dimension = number of vectors in a basis.
Dimension is the effective number of degree of freedom
For any vector/sub-spaces, you can have different basis but the dimension is always the same, i.e. a unique number.
Properties of basis and dimensions:

Dimension is the maximum number of lin ind vectors

Dimension is the minimum number of vectors that can span the whole space

In any n-dim space, any n lin ind vectors must span,

In any n-dim space, any n vectors that span must be lin ind,

In any n-dim space, for any n vectors: linear independence <=> span <=> basis.

All of the above can be understood by means of the following two theorems:
consider AX=b, A is an mxn matrix:
More Unknown Theorem (n > m): there must be some free variables; for consistent system, there must be infinitely many solutions;
(i.e., the coloumns of the matrix A must be linearly dependent.)
More Equation Theorem (m > n): there must be some b such that AX=b has no solution.
(i.e., the coloumns of the matrix A cannot span the whole R^m.)

Note: Basis and Dimension

Homework 5, due: Thursday, Oct 23rd, in class.

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Week 8 (October Break, Oct 16):

(Continuation on basis and dimensions.)
Two "generic" methods to find basis:
Method 1: given a list of spanning vectors, throw away all the redundant vectors.
(Typical example: find a basis for Col(A).)
Method 2: express (linear) condition/constraint in terms of free variables.
(Typical example: find a basis for Null(A).)
(You need to find "A" for each individual problem.)

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Week 9 (Oct 21, 23):

[P 2.3] Col, Null, and Row spaces associated with a matrix.
Row(A)=Span of rows of A;
Row space is preserved under elementary row operations.
dim(Col) = number of pivots = Rank;
dim(Null) = number of free varables = Nullity;
dim(Row) = number of non-zero rows in (R)REF = number of pivots = Rank;
dim(Col)=dim(Row)=Rank.
Rank-Nullity Theorem: Rank + Nullity = Total number of variables.
Relationship between rank and nullity with linear independence: (A, mxn matrix)

column of A are lin. ind. <=> Rank(A)=n <=> Nullity(A)=0
Relationship between rank and nullity with solvability and uniqueness of solution: (A, mxn matrix)

solution of AX=b (if exists) is unique <=> Nullity(A) = 0 <=> Rank(A)=n.

AX=b is always solvable (for any b) <=> Rank(A)=m.
[P 1.4] Subspaces
closed under vector addition and scalar multiplication

Note: Col, Null, and Row spaces

Homework 6, due: Thursday, Oct 30th, in class.

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Week 10 (Oct 28, 30):

[P 3.1, 3.2]
Linear transformations and their matrix representations
Geometric examples of linear transformations: projection, reflection, rotation;
Composition of linear transformations and matrix multiplication: [TS]=[T][S]
Beware of dimension compatibility: C^(mxn) = A^(mxl)*B^(lxn)
Properties of matrix multiplications: distributive, associative.
In general, AB is not equal to BA
Diagonal matrices, identity matrices.

Note: Matrix Multiplication

Homework 7, due: Friday, Nov 7th, 12:00pm, in MATH 432 (slide under the dooor, if closed).

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Week 11 (Nov 4, 6):

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Week 12 (Nov 11, 13):
Midterm Two: in class, Thursday, Nov 13

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Week 13 (Nov 18, 20):

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Week 14 (Nov 25, Thanksgiving):

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Week 15 (Dec 2, 4):

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Week 16 (Dec 9, 11):

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Week 17 (Final Exam Week):
Tuesday, Dec 16, 10:30am-12:30pm, LILY 3118