Linear Algebra II

Purpose of Course  showclose

Linear algebra is the study of vector spaces and linear mappings between them.  In this course, we will begin by reviewing topics you learned in Linear Algebra I, starting with linear equations, followed by a review of vectors and matrices in the context of linear equations.  The review will refresh your knowledge of the fundamentals of vectors and of matrix theory, how to perform operations on matrices, and how to solve systems of equations.  After the review, you should be able to understand complex numbers from algebraic and geometric viewpoints to the fundamental theorem of algebra.  Next, we will focus on eigenvalues and eigenvectors.  Today, these have applications in such diverse fields as computer science (Google’s PageRank algorithm), physics (quantum mechanics, vibration analysis, etc.), economics (equilibrium states of Markov models), and more.  We will end with the spectral theorem, which provides a decomposition of the vector space on which operators act, and singular-value decomposition, which is a generalization of the spectral theorem to arbitrary matrices. Then, we will study vector spaces: real, complex, and abstract (i.e., vector space of dimension N over an arbitrary field K) linear transformations.  Vector spaces are structures formed by a collection of vectors and are characterized by their dimensions.  We will then introduce a new structure on vector spaces: an inner product.  Inner products allow us to introduce geometric aspects, such as length of a vector, and to define the notion of orthogonality between vectors.  In this context, we will study the geometric aspects of linear algebra by using Euclidean spaces as a guide.  If you encounter a theorem that seems difficult or does not seem intuitive, try to study that theorem in the simplest case possible and then move on to more abstract cases.  For example, if you are uncomfortable with abstract vector spaces (V) over an arbitrary field (K), then you can fall back on intuition from such spaces as R and C (real and complex). Alternatively, you can reduce the dimension of the vector spaces involved as many notions can be understood in the two-dimensional case.

Course Information  showclose

Welcome to MA212.  Below, please find general information on the course and its requirements.  
Primary Resources: This course is comprised of the following primary materials:

Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials.  You will also need to complete the following:
- Sub-subunit 1.1.3 Activity
- Sub-subunit 1.1.4 Activity
- Subunit 1.2 Activity
- Sub-subunit 1.3.1 Activity
- Sub-subunit 1.3.2 Activity
- Subunit 1.4 Activity
- Sub-subunit 2.1.2 Activities
- Subunit 2.3 Activity
- Subunit 2.4 Activity
- Sub-subunit Activity
- Sub-subunit Activity
- Sub-subunit 3.1.3 Activity
- Sub-subunit 3.2.3 Activities
- Subunit 3.3 Activity
- Subunit 3.4 Activity
- Sub-subunit 4.1.3 Activities
- Sub-subunit 4.2.4 Activities
- The Final Exam
Note that you will only receive an official grade on your Final Exam.  However, in order to adequately prepare for this exam, you will need to work through the resources in each unit and the activities listed above.
In order to “pass” this course, you will need to earn a 70% or higher on the Final Exam.  Your score on the exam will be tabulated as soon as you complete it.  If you do not pass the exam, you may take it again.
Time Commitment: This course should take you a total of 117.5 hours to complete.  This is only an approximation, and the course may take longer to complete.  Each unit includes a “time advisory” that lists the amount of time you are expected to spend on each subunit.  These should help you plan your time accordingly.  It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself.  For example, Unit 1 should take you about 32.5 hours to complete.  Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 11.5 hours) over three days, for example by completing sub-subunits 1.1.1, 1.1.2, and half of 1.1.3 (a total of 4.5 hours) on Monday; the second half of sub-subunit 1.1.3 and sub-subunit 1.1.4 (a total of 5 hours) on Tuesday; and sub-subunit 1.1.5 and 1.1.6 (about 2 hours) on Wednesday; etc.
Tips/Suggestions: As noted in the “Course Requirements,” Linear Algebra I is a pre-requisite for this course.  If you are struggling with the material as you progress through this course, consider taking a break to revisit  MA211 Linear Algebra.  It will likely be helpful to have a graphing calculator on hand for this course.  If you do not own or have access to one, consider using this free graphing calculator.  As you read, take careful notes on a separate sheet of paper.  Mark down any important equations, formulas, and definitions that stand out to you.   These notes will serve as a useful review as you study for the Final Exam.

A version of this course is also available in iTunes U.
Preview the course in your browser or view all our iTunes U courses.  

Learning Outcomes  showclose

Upon successful completion of this course, the student will be able to:
  • Define and apply the abstract notions of vector space and inner product space.
  • Identify examples of vector spaces.
  • Diagonalize a matrix.
  • Formulate what a system of linear equations is in terms of matrices.
  • Determine whether or not a space has the Archimedean property.
  • Use the polar form and geometric interpretation of the complex numbers to solve problems.
  • Explain what the fundamental theorem of algebra states.
  • State the Fredholm alternative.
  • Compute eigenvalues and eigenvectors.
  • State Schur's Theorem.
  • Define normal matrices.
  • Explain the composition and the inversion of permutations.
  • Define and compute the determinant.
  • Explain when eigenvalues exist for a given operator.
  • Compute the normal form of a nilpotent operator.
  • Explain the idea of Jordan blocks, Jordan matrices, and the Jordan form of a matrix.
  • Define and identify quadratic forms.
  • State the second derivative test.
  • Define and compute eigenvectors and eigenvalues.
  • Define a vector space and state its properties.
  • Compute the linear span of a set of vectors.
  • Determine the linear independence or dependence of a set of vectors.
  • Determine a basis of a vector space.
  • Explain the ideas of linear independence, spanning set, basis, and dimension.
  • Define and identify linear transformations.
  • Define and compute the characteristic polynomial of a matrix.
  • Define and compute a Markov matrix.
  • Identify and compute stochastic matrices.
  • Define and identify normed vector spaces.  
  • Apply the Cauchy Schwarz inequality.
  • State the Riesz representation theorem.
  • State what it means for a nxn matrix to be diagonalizable.
  • Define and identify Hermitian operators.
  • Define and identify Hilbert spaces.
  • Prove the Cayley-Hamilton theorem.
  • Define and determine the adjoint of an operator.
  • Define and identify normal operators.
  • State the spectral theorem.
  • Explain how to find the singular-value decomposition of an operator.
  • Define the notion of length for abstract vectors in abstract vector spaces.
  • Define and identify orthogonal vectors.
  • Define and identify orthogonal and orthonormal subsets of R^n.
  • Perform range-nullspace decompositions.
  • Perform orthogonal decomposition of space.
  • Perform singular-value decomposition.
  • Use the Gram-Schmidt process.
  • Read and interpret advanced mathematical definitions, theorems, and proofs.
  • Produce well-written proofs.

Course Requirements  showclose

In order to take this course, you must:
√    Have access to a computer.
√    Have continuous broadband Internet access.
√    Have the ability/permission to install plug-ins or software (e.g., Adobe Reader or Flash).
√    Have the ability to download and save files and documents to a computer.
√    Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).
√    Be competent in the English language.
√    Have read the Saylor Student Handbook.
√    Have taken the following course from the “Core Program” of the mathematics discipline as a pre-requisite: MA211: Linear Algebra.

Preliminary Information

  • Preliminary Information

    Open Textbook Challenge Winner: Linear Algebra, Theory and Applications

    Linear Algebra, Theory and Applications was written and submitted by Dr. Kenneth Kuttler of Brigham Young University.  Dr. Kuttler wrote this textbook for use by his students at BYU.  According to the preface of the text, “This is a book on linear algebra and matrix theory.  While it is self-contained, it will work best for those who have already had some exposure to linear algebra.  It is also assumed that the reader has had calculus.  Some optional topics require more analysis than this, however.”  A solutions manual to the textbook is included.

    Linear Algebra, Theory and Applications (PDF)
    Linear Algebra, Theory and Applications Solutions Manual (PDF)

Unit Outline show close

Expand All Resources Collapse All Resources