MA 351: Elementary Linear Algebra
Spring 2019, Purdue University

Course Description:

Systems of linear equations, matrices, finite dimensional vector spaces, determinants, eigenvalues and eigenvectors, inner products and orthogonality


Aaron Nung Kwan Yip
Department of Mathematics
Purdue University

Contact Information:

Office: MATH 432
Email: click here

Lecture Times and Places:

Section 112 (CRN 15967): T, Th 09:00 - 10:15, UNIV 217

Office Hours:

Tue: 2:00-3:00pm, Wed: 4:00-5:00pm, 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.

    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:

    University policy and procedures will be followed.
    For more detail information, please click ADA Information

    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;
    Chapter 6: orthogonality

    Course Progress and Announcement:

    (You should consult this section regularly, for homework assignments, additional materials and announcements.)

    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:

    Tues, Jan 8:
    [P 1.2] Geometric interpretations of finding solutions:
    (i) (row) intersection between lines, planes;
    (ii) (column) writing vector as linear combination.
    NOTE: Two interpretations of solving linear systems

    Thur, Jan 10:
    [P 1.2] Elementary row operations; three possibilities of scenarios:
    (i) unique solution; (ii) infinitely many solutions; (iii) no solutions.

    Homework 1: due Thursday, Jan 17, in class.
    (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:

    Tues, Jan 15:
    [P 1.3] Elementary row operations, Gaussian elimination.
    NOTE: Examples on solving mxn systems
    (m - number of equations - might not equal n - number of unknowns)

    Thur, Jan 17:
    [P 1.3] Key concepts coming out 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
    Some applications: interpolating polynomials, traffic flows.

    Homework 2: due Thursday, Jan 24, in class.
    Homework 2

    Week 3:

    Tues, Jan 22:
    [P 1.1] Vectors in R^n:
    vector addition, scalar multiplication, and their properties;
    linear combinations and span.
    NOTE: (General) Vector Space

    Thur, Jan 24:
    [P 1.1] polynomials, functions, matrices as vector spaces;
    [P 1.4] matrix multiplied by a column vector, NOTATION MATTERS:
    A(X+Y) = AX + AY; A(aX) = aAX; (A+B)X = AX + BX;
    linear system of equation in matrix form: AX=b;
    Column space of A, Col(A): AX=b is solvable if and only if b belongs to Col(A).

    Homework 3: due Thursday, Jan 31, in class.
    p.17: #3, 5, 26;
    p.86: #96, 97, 106, 107, 108, 109, 111, 112, 121.

    Week 4:

    Tues, Jan 29:
    [P 1.4] homogeneous (AX=0) vs inhomogeneous (AX=b) systems;
    Null space of a matrix A (Null(A)): solution of the homogeneous system.
    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

    [P 1.1] linear dependence and redundant vectors:
    how to determine if a vector is redundant.

    Thur, Jan 31:
    [P 2.1] linear dependence and independence;
    How to eliminate all redundant vectors.
    NOTE: Linear Dependence and Linear Independence

    Homework 4: due Thursday, Feb 7, in class.
    p.108: #2.1, 2.3, 2.7, 2.11, 2.13, 2.17, 2.18.

    Week 6:

    Test One: Feb 14, in class.
    Materials covered: Chapter 1 to Chapter 2.2.
    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.
    No calculator or any electronic devices are allowed (nor needed.)
    Past Exam One
    Solution of Quiz 1-5
    Solution of Hw 1, 2, 3, and 4

    Test One Statistics: (Solution)
    Total number of students = 45
    A (80 <= scores <= 100): No. of students = 15 (33%)
    B (60 <= scores <= 79): No. of students = 16 (36%)
    C (40 <= scores <= 59): No. of students = 9 (20%)
    D (20 <= scores <= 39): No. of students = 4 (9%)
    F (scores <= 19): No. of students = 1 (2%)
    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.

    Homework 7: due Thursday, Feb 21, in class.
    Homework 5

    Week 7:

    Tue, Feb 19:
    [P 2.1, 2.2] NOTE: Basis and Dimension
    More unknown theorem:
    In any linear system with m equations and n unknowns with n > m, there must be at least one free variables.
    More equation theorem:
    In any linear system with m equations and n unknowns with m > n, there must be a vector B such that AX=B is not solvable.
    Well-definedness (uniqueness of) dimension;
    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

    Thur, Feb 21:
    [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 1.4] Subspaces
    Subspace is closed under vector addition and scalar multiplication

    NOTE: Col, Null and Row spaces

    Homework 6: due Thursday, Feb 28, in class.
    p.143: #2.64, 2.65, 2.66(a,c), 2.67(a,c), 2.76, 2.78
    p.92: #1.115, 1.116, 1.117, 1.118, 1.119, 1.120

    Week 8:

    Tue, Feb 26:
    [P 3.1] Linearity properties:
    closed under vector addition and scalar multiplication
    definition of subspaces, matrix multiplication (X-->AX),
    Linear transformations (X--> T(X)), examples: reflection, projection, rotations

    Thur, Feb 28:
    [P 3.1] 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.

    Homework 7: due Thursday, Mar 7, in class.
    p.158: #3.1, 3.2, 3.5, 3.6, 3.10, 3.11, 3.12, 3.13, 3.15
    p.173: #3.26, 3.27
    Week 9:

    Tue Mar 5, Thur Mar 7:
    [P 3.2] Matrix multiplications
    C^(mxn) = A^(mxl)*B^(lxn)
    beware of dimension compatibility
    connection to composition of linear transformation: [TS]=[T][S]
    In general, AB is not equal to BA

    Homework 8: due Thursday, Mar 21, in class.
    p.174: #3.34, 3.35, 3.36, 3.37, 3.41, 3.42, 3.44, 3.48, 3.49, 3.50, 3.52
    (hint for 3.48, 3.49, 3.50: look at 3.52)
    Additional problem #1: find the matrix P that projects vectors in R^2 onto the straight line y=mx. Show that P^2 = P.
    Additional problem #2: find the matrix R that reflects vectors in R^2 with respect to the straight line y=mx. Show that R^2 = I.

    (Week 10: Spring Break)
    Week 11:

    Tue Mar 19:
    General theory of maps and functions,
    onto (surjective) and one-to-one (injective) maps,
    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;
    A^(mxn) is one-to-one <=> AX=Y has unique solution (if solvable) <=> A has no free variables <=> Rank(A) = n

    Thur Mar 21
    [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)
    finding A inverse by row operations
    properties of inverse

    NOTE: Inverse of a matrix
    NOTE: Two applications of matrix: Leontief input-output model and graph theory
    Leontief, Input Out Economics, Scientific American, 1951

    Homework 9: due Thursday, Mar 28, in class.
    p. 190: #3.64(a to j), 3.72, 3.74, 3.75, 3.76, 3.77, 3.78, 3.87
    p. 202: (Self-Study Questions) 3.4, 3.5(a,b)
    For the graph (Figure 3.10) in p. 180, the connectivity matrix M and M^2 are as given in the text. Find also M^3.
    Using M^2 and M^3, indicate:
    (1) how many 2-step paths are there from D to B and D to D?
    Draw these paths explicitly, each on a separate graph.
    (2) how many 3-step paths are there from B to B and B to D?
    Draw these paths explicitly, each on a separate graph.

    Week 12:

    Tue Mar 26:
    [P 4.1, 4.2] Determinant of a square matrix
    (I) computation using co-factor expansion;
    (II) computation using row reduction.

    Thur Mar 28:
    [P 4.2, 4.3]
    Properties of determinants
    Applications of determinants:
    computation of area of parallelogram and volume of parallelipiped
    solution of AX=B (for invertible, square matrix A) (Cramer's Rule)

    Week 13:

    Test Two: Apr 4th, in class
    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.
    No calculator or any electronic devices are allowed (nor needed.)
    Past Exam Two
    Solution of Quiz 6
    Solution of Quiz 7
    Solution of Hw 5, 6, 7, 8, and 9

    Test Two Statistics: (Solution)
    (Total number of students = 45)
    A (80 <= scores <= 100): No. of students = 19 (42%) (Test One: 15, 33%)
    B (60 <= scores <= 79): No. of students = 15 (33%) (Test One: 16, 36%)
    C (40 <= scores <= 59): No. of students = 8 (18%) (Test One: 9, 20%)
    D (20 <= scores <= 39): No. of students = 3 (7%) (Test One: 4, 9%)
    F (scores <= 19): No. of students = 0 (0%) (Test One: 1, 2%)
    (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.)

    Homework 10. Due: Apr. 11, Thursday, in class
    p.249, #4.1
    p.259, #4.12, 4.15, 4.16, 4.17, 4.24, 4.25, 4.26
    p.268, #4.34, 4.36, 4.41

    Week 14:

    Tue Apr 9:
    [P 4.2, 4.3] Properties of determinants and their applications.
    Cramer's rule for solving AX=b (for invertible square matrices)
    Formula for A^(-1) (if det(A) not equal to zero).

    [P 5.1] Eigenvalue and eigenvectors:
    AX=lambda X (X must not be the zero vector)

    Thur Apr 11:
    [P 5.1] Eigenvalues and eigenvectors: AX=lamba X
    Examples with distinct and repeated eigenvalues
    Applications: computing (A^n)Y

    Homework 11. Due: Apr. 18, Thursday, in class
    p. 280: #5.3, 5.5, 5.10, 5.11, 5.12, 5.13, 5.14, 5.15, 5.16

    Week 15:

    Tue Apr 16:
    [P 5.1] distinct and repeated eigenvalues
    Algebraic multiplities (m_i) vs geometric multiplicities (g_i),
    deficient/defective eigenvalues and matrices,
    independence of eigenvectors with distinct eigenvalues,
    an application: matrix powers and Fibonacci sequence.

    Thur Apr 18:
    [P 5.2] Diagonalizable matrices and diagonalization process
    [P 5.3] Complex eigenvalues and eigenvectors

    Week 16:

    Tue Apr 23
    [P 5.3] complex numbers;
    NOTE: Complex Numbers
    [p 5.12] application to Markov processes

    Thur Apr 25
    An overview of the course

    Homework 12. For practice. No need to hand in.
    p. 288: #5.17, 5.20, 5.21;
    p. 290: #5.27, 5.28, 5.29, 5.31, 5.32, 5.33, 5.34, 5.35, 5.37, 5.38, 5.39;
    p. 304: #5.44, 5.45, 5.46, 5.47, 5.48, 5.49

    Final Exam: Thur, 05/02, 8am-10am, SC 239

    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.)

    A past final exam
    Solution of Hw 10, 11, 12
    Solution of Quiz 8
    Solution of Quiz 9

    Office hours during exam week:
    Monday 3:30-5:30, Tuesday 4:30-6, Wednesday 5-6

    Solution of final exam