MA 351: Elementary Linear Algebra
Fall 2021, Purdue University

Course Description:

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


Aaron Nung Kwan Yip
Department of Mathematics
Purdue University

Contact Information:

Office: MATH 432
Email and Phone: click here

Lecture Times and Places:

Section 022 (CRN 64933): T, Th 10:30 - 11:45, UNIV 117

Office Hours:

Tue: 2:00-3:15pm, Wed: 3:00-4:15pm, or by appointment.


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.


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.

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

  • Please resolve any error in the grading (hws and tests) WITHIN ONE WEEK after the return of each homework and exam.

  • No late homeworks are 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 and Midterm Two, in class, dates TBA

    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.

    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 welcome to let me know so that we can discuss options. You are also encouraged to contact the Disability Resource Center at: or by phone at 765-494-1247.

    If you have been certified by the Disability Resource Center (DRC) as eligible for accommodations, you should contact me to discuss your accommodations as soon as possible. Click here for instructions for sending your Course Accessibility Letter to me.

    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.

    Protect Purdue Plan:

    Protect Purdue Plan / (Pledge)
    Essential Protect Purdue Guidance (and frequently asked questions)
    Compliance Plan: What to do in the presence of violations: Ask, Offer, Leave, Report

    Lack of compliance
    Students who are not engaging in behaviors established in the standard operating procedures (e.g., properly wearing a mask when required) will be asked to comply and offered any assistance they need in order to comply. If non-compliance continues, possible results include instructors asking students to leave the class, potentially followed by instructors dismissing the whole class. Students who do not comply with the required health and Protect Purdue Pledge behaviors are violating the University Code of Conduct and will be reported to the Dean of Students Office, with sanctions ranging from educational requirements to dismissal from the university. For additional guidance, please see the Dean of Students guidance on Managing Classroom Behavior and Expectations.

    Student rights
    Any student who has substantial reason to believe that another person in the room is threatening class safety by not wearing a face covering or following other safety guidelines for public health considerations may leave the class without consequence. The student is encouraged to report the observed behavior to the course instructor or to the Office of Student Rights and Responsibilities (OSRR), as well as discuss next steps with the instructor.

    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.

    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 concepts of abstract (and yet concrete and useful) vector spaces, in particular, basis, dimension, and geometry of subspaces;
    (4) last but not least, understanding and appreciation of the need of giving proofs, how to write proofs and knowing what constitutes a proof.

    Reading of the textbook. I highly encourage you to read the textbook.
    I will try to follow the materials from the textbook, but inevitably there will be deviations: in the presentation styles, emphasis, examples, and solution methods. Going to the lectures and reading the textbook can thus give you multiple viewpoints on the materials. In addition, the textbook has many worked out examples.

    Use of technology. You are highly encouraged to try and experiment with technology, in particular Matlab. (I often use it myself.)
    My motto on the use of technology:
    If technology helps you understand, by all means use it. Otherwise, use it at your own risk.
    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.

    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.

    Week 1 (Aug 24, 26):

    [P 1.2]
    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 (include solution of Quiz #1)

    Homework 1: due Tuesday, Sept 7, in class. (note: new due date)
    (Unless otherwise stated, all homework problems are from the textbook, Penney, Linear Algebra, 4th Edition.)
    p.39: #55, 59;
    p.64: #67, 69.

    Week 2 (Aug 31, Sept 2):

    [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)

    Note: Gaussian Elimination
    Note: Examples on solving mxn systems

    Homework 2: due Thursday, Sept 9, in class.
    p.63: #65, 74, 75, 78, 79, 80.

    Week 3 (Sept 7, 9):

    Some applications of solving mxn linear systems:
  • expressing vectors as linear combinations [P 1.1];
  • interpolating polynomials;
  • traffic flows [P 1.3.2].

    [P 1.1] Vectors in R^n:
  • vector addition, scalar multiplication, and their properties;
  • linear combinations and span and their interpretation in terms of solving linear system.

    Note (revised): Concept of a general Vector Space

    Homework 3: due Thursday, Sept 16, in class.
    Click here

    Week 4 (Sept 14, 16):

    [P 1.1] Polynomials, functions, matrices as vector spaces;
    [P 1.1] Linear dependence and linear independence.
    Linear dependence: linear relations between the vectors.
    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 matrix A, Null(A)={X: AX=0}.
    Homogeneous (AX=0) vs inhomogeneous (AX=b) systems;
  • AX=b is solvable if and only if (iff) b belongs to Col(A).
  • Structure of solutions for AX=b (assume it is consistent):
    X = p + Null(A), where p is a translation vector, or a particular solution.
  • Suppose AX=b is solvable. Then solution is unique if and only if (iff)
    Null(A) = {0}
    , i.e. there is no free variables, i.e. all variables are pivots.

    Note: Lecture of Sept 16
    Note (revised): Column and Null spaces of a matrix

    Homework 4: due Thursday, Sept 23, in class.
    p.86: #96, 97, 106, 107, 108, 109, 111, 112, 121.

    Week 5 (Sept 21, 23):

    How to check if two spans or solution representations are the same.
    [P 2.1] linear dependence and independence.
    Dependence relation/equation,
    Consequence of linear dependence.
    Redundant vectors, how to throw away all the redundant vectors.
    Basis vectors = spanning vectors with no redundancy.

    Note: Linear Dependence and Independence

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

    Week 6 (Sept 28, 30):

    Midterm 1: Thursday, Sept 30, in class.
    No calculator nor any electronic devices are allowed (nor needed.)
    Materials covered: Penney, Chapter 1 to Chapter 2.1.
    The best way to review is to:
    (i) go over lecture materials,
    (ii) read the textbook, and
    (iii) go over the homework and quiz problems.

    Solution of Quiz 1-4
    Selected Solution of Hw 1, 2, 3, 4, and Problems for [P 2.1]
    Past Exam One (Spring 2016)
    Past Exam One (Spring 2019)

    Test One Statistics: Total number of students = 38
    A (74 <= scores <= 100): No. of students = 13 (34%)
    B (54 <= scores <= 73): No. of students = 9 (24%)
    C (30 <= scores <= 53): No. of students = 12 (32%)
    D (20 <= scores <= 29): No. of students = 2 (5%)
    F (scores <= 19): No. of students = 2 (5%)
    Average = 61
    Min = 21, Max = 99
    Mode = 75
    Median = 65
    Standard Deviation = 20
    (Note: the above cut-offs are very rough and simple cut-offs, purely based on the test scores. I have not considered the hws, and quizzes.)

    Test 1 Solution

    Week 7 (Oct 5, 7):

    [P 2.1]
    Basis vectors for a vector space V:
    (i) span the whole V, i.e. any vector from V can be written as a linear combination of the basis vector;
    (ii) linear independent, i.e. no redundant vectors.
    How to find a 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).)
    [P 2.2]
    Dimension of a vector space.

    Homework 5: due Thursday, Oct 7, in class.
    Click here

    Homework 6: due Thursday, Oct 21, in class.
    Click here

    Week 8 (Oct 14):

    [P 2.1, 2.2]
    More unknown theorem:
    In any linear system with m equations and n unknowns, if n > m, there must be at least one free variable.
    More equation theorem:
    In any linear system with m equations and n unknowns, if m > n, there must be a vector B such that AX=B is not solvable.
    Even though there can be many basis of the same space, the dimension is unique.
    Properties of dim:
  • Dimension is the maximum number of lin ind vectors
  • Dimension is the minimum number of vectors that can span
  • 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,
    Dimension is the effective number of degree of freedom in a vector space.

    Note: Basis and Dimension

    Week 9 (Oct 19, 21):

    [P 2.3]
    Col, Null, and Row spaces associated with a matrix.
    dim(Col) = number of pivots = Rank;
    dim(Null) = number of free varables = Nullity;
    Rank-Nullity Theorem: Rank + Nullity = Total number of variables.
    Relationship between rank and nullity with lineary independence;
    Relationship between rank and nullity with solvability and uniqueness of solution
    Non-singular matrix A: AX=B has a unique solution for any B.
    Equivalent properties of non-singular matrices.
    [P 1.4]
    Subspaces is closed under vector addition and scalar multiplication

    Note: Col, Null and Row spaces

    Homework 7: due Thursday, Oct 28, in class.
    Click here

    Week 10 (Oct 26, 28):

    [P 3.1] Linearity properties: closed under vector addition and scalar multiplication
    Definition of subspaces, examples
    Matrix multiplication by a vector: X-->AX
    Linear transformations: X--> T(X)
    Examples: reflection, projection, rotations
    Linear transformations given by matrix multiplications:
    T(X) = AX: X (in R^n) ---> AX (in R^m), where A is an mxn matrix.
    Matrix representation of T, how to find the matrix corresponding to T.
    [P 3.2] Matrix multiplications
    C^(mxn) = A^(mxl)*B^(lxn), beware of dimension compatibility

    Homework 8: due Thursday, Nov 4, in class.
    p.158: #3.1, 3.2, 3.5, 3.6, 3.10, 3.11, 3.12, 3.15
    p.174: #3.41, 3.44
    p.190: #3.64(a to j), 3.65(a to j), 3.72

    Week 11 (Nov 2, 4):

    [P 3.2] Matrix multiplications
    Connection to composition of linear transformation: [TS]=[T][S]
    In general, AB is not equal to BA

    General theory of functions (or maps):
    onto (surjective), one-to-one (injective), inverse of a map (for onto and one-to-one maps)
    A^(mxn) is onto
    <=> AX=Y is solvable for any Y
    <=> A has m pivot (the maximum number) variables
    <=> Rank(A)=m, i.e. A has full rank;
    A^(mxn) is one-to-one
    <=> AX=Y has unique solution (if solvable)
    <=> A has no free variables
    <=> Nullity(A) = 0
    <=> Rank(A) = n

    [P 3.3] Inverse of a matrix
    A has an inverse
    <=> A is onto and one-to-one
    <=> AX=Y is uniquely solvable for any Y
    <=> Rank(A)=m=n (hence A is necessarily a square matrix)
    (A is non-singular.)
    Finding the inverse of A by row operations:
    If [A | I] -- (row reduction) -->[I | B], then B = A^(-1).
    Properties of inverse: AA^(-1) = A^(-1)A = I.

    Note: Inverse of a matrix

    Week 12 (Nov 9, 11):

    Midterm 2: Thursday, Nov 11, in class.
    No calculator or any electronic devices are allowed (nor needed.)
    Materials covered: Chapter 2 to Chapter 3. (Note: The whole Chapter 2 is included.)
    (Even though I will not specifically ask questions about the materials before Test One, I do not know of any concepts before Test One that will not be used for Test Two.)
    The best way to review is to:
    (i) go over lecture materials,
    (ii) read the textbook, and
    (iii) go over the homework and quiz problems.

    Past Exam Two (Spring 2016)
    Past Exam Two (Spring 2019)

    Note: Review Lecture (Tue Nov 9), Review Lecture (Wed Nov 10)
    (Go to Brightspace, Content/Course Information to view both Tue and Wed's recorded review lecture.)

    Test Two Statistics: Total number of students = 34
    A (75 <= scores <= 100): No. of students = 12 (35%)
    B (55 <= scores <= 74): No. of students = 13 (38%)
    C (35 <= scores <= 54): No. of students = 9 (26%)
    Average = 69 (Test One: 61)
    Min = 40 (21), Max = 100 (99)
    Mode = 82 (75)
    Median = 66 (65)
    Standard Deviation = 18 (20)
    (Note: the above cut-offs are very rough and simple cut-offs, purely based on the test scores. I have not considered the hws, and quizzes.)

    Test 2 Solution
    (Go to Brightspace, Content/Course Information for recorded comments of Test Two Solution.)

    Week 13 (Nov 16, 18):

    Applications of matrices: rotation, projection and reflection.
    [P 4.1, 4.2, 4.3] Determinant of a square matrix
    (I) computation using co-factor expansion along any row (or column);
    (II) computation using row (or column) reduction.
    Applications of determinants:
  • computation of area of parallelogram and volume of parallelepiped
  • Cramer's rule for solving AX=b (for invertible square matrices)
  • formula for matrix inverse

    Week 14 (Nov 23):

    Further properties and applications of determinants.

    Note: Lecture of Nov 23
    (Go to Brightspace, Content/Course Information for the recorded lecture.)

    Homework 9: due Thursday, Dec 2, in class.
    p.176: 3.51, 3.52
    p.192: 3.74, 3.75, 3.76, 3.77, 3.78, 3.85, 3.87
    p.249: 4.1
    p.259: 4.12, 4.15, 4.16, 4.17, 4.19, 4.24, 4.25
    p.268: 4.34, 4.36, 4.41

    Week 15 (Nov 30, Dec 2):

    [P 5.1] Eigenvalue and eigenvectors:
    AX=lambda X: X must not be the zero vector.
    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 with distinct eigenvalues

    Note: Eigenvalues and Eigenvectors

    Week 16 (Dec 7, Dec 9):

    [P 5.1, 5.2]
    An application of eigenvalues and eigenvectors: computing matrix powers.
    Use eigenvectors as basis.
    Diagonalizable matrices and diagonalization process
    Application to Markov processes.

    Practice problems for Chapter 5 (no need to hand in):
    p. 280: #5.3, 5.5, 5.10, 5.11, 5.12, 5.13, 5.14, 5.15, 5.16;
    p. 285: #5.17, 5.20;
    p. 290: #5.27, 5.28, 5.31, 5.32, 5.33, 5.34, 5.35, 5.37, 5.38, 5.39.

    Note: Summary of 351

    Final Exam: Wed 12/15, 8:00am - 10:00am, BRNG 2290

    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 the homework and quiz problems and understand.
    No calculator or any electronic devices are allowed (nor needed).

    Past final (Spring 2016)
    Past final (Spring 2019)

    Help sessions in zoom: Sunday, Monday 5pm-6pm.
    (Both sessions will be recorded and posted in Brightspace.)

    Final Review - Part 1 (Dec 9)
    Final Review - Part 2 (Dec 12)
    Final Review - Part 3 (Dec 13)
    Final Review - Part 4 (Dec 14)
    (Go to Brightspace for recorded lecture.)