• Syllabus
  • Text: Linear Algebra by Jim Hefferon
  • Companion Notes Site: OER – Western CT Linear Algebra
  • Class Calendar: Your syllabus has a rough calendar of what we will be covering each class.  After each class I will post what we were actually were able to cover here:
    1. 8/30/2018 – We went over the syllabus and then learned about some matrix and vector operations in the Fall 2018 Opener, try and finish this for next Thursday.
    2. 9/6/2018 – We covered solving systems using elimination and back substitution and covered which types of systems of equations we may encounter.  At the end of class we introduced the idea that we can really study all of this by just looking at the coefficients and constants.  This material goes with Chapter 1 Section 1 in OER -WCSU Linear Notes, Chapter 1 Section 1 in your Text, and here are notes from one of your class mates.
    3. 9/10/2018 – We introduced Row Operations, Row Echelon Form, Row Reduced Echelon Form, the idea of General/Parametric Solutions to systems, and Zero Divisors.  We also touched on the connection between matrix operations and solving equations so that we can view solving systems to be like solving equations of the form \(A\vec{x}=\vec{y}\) or \(A\vec{x}=\vec{0}\). This material goes with Chapter 1 Section 1 in OER -WCSU Linear Notes and  Chapter 1 Section 1 in your Text.
    4. 9/13/2018: Today we practiced row operations and started looking at the relationship between algebraic and geometric ideas.  So far we have covered most of the content in Chapter 1 Sections 1 and 3 in your Text, all of Chapter 1 Section 1 in OER -WCSU Linear Notes (hinting at material in sections 2 and 3 as well), and here are notes written up by one of your classmates: Notes from 9-13-2018
    5. 9/17/2018 – Today we talked about writing systems of equations in vector form so that we can understand them geometrically, these ideas are touched on in the online notes in chapter 1 section 2.  This is a perspective that will become really important when we start to learn about vector spaces and projections.  Please try to read through the pp.35-42 in your textbook and chapter 1 section 2 in the online notes.
    6. 9/20/2018: Today we discussed the different perspectives from which we will be approaching material this semester.  We also learned about the dot product, how to calculate it and how to think about it geometrically.  Look at the notes here and here, and at the text here. And, here are some notes from a classmate: Notes from 9-20-2018
    7. 9/24/2018: Your first exam was today.
    8. 9/27/2019: We went over the definition of a vector space and looked at examples.  We also introduced the idea of a basis.  This material is here in the online notes, and here in your text.
    9. 10/1/2018: We spent more time on vector spaces covering the definitions of: basis, dimension, linearly independent, linearly dependent, and subspace. This material is here in the online notes, and here in your text.
    10. 10/4/2018: Today we introduced the definition of span.  At the end of class we were playing with the system defined by \(f(x)=ax^2+bx+c\) with the conditions \(f(1)=0\) and \(f(2)=0\), we will pick up there on Monday.  Finally, I am moving the exam from next Thursday, the 11th, to the following Monday the 15th.  Please take this opportunity to look at more practice problems. Here is the most relevant part of the online notes and here is the appropriate place in the text for today’s work.
    11. 10/8/2018: We covered the Row, Column, and Null Spaces.  Be sure to study the unit vocabulary, I updated it below.  Also, check out the extra credit opportunities listed below.
    12. 10/11/2018: We talked more about row, column, and null spaces ahead of the exam on Monday.  Information on these can be found here in your text and here in the online notes. And here are notes form a classmate.
    13. 10/15/2018: Exam 2
    14. 10/18/2018: We started talking about linear transformations, introducing the terms function, linear transformation, one-to-one, onto, domain, codomain, range (image), and kernel (null space).  The corresponding place in the online notes is here, and in your text you need to look here (pp. 165-203). And here are notes from a class mate.
    15. 10/22/2018: We practiced showing that a transformation was linear (study your definitions!!), we introduced the idea of writing linear transformations as matrices, and ended by looking at how we can use (or at least try to use) row operations to find the inverse of a matrix, along the way we saw that row operations can also be written as matrix products.  In the online notes you should look here for notes transformations and here for notes on matrix operations, in the text you should be looking at the pages 165-253 for material on transformations and matrix operations.
    16. 10/25/2018: We practiced finding matrix inverses, discussed the relationship between the Domain and the Row Space and Null Space (kernel) and between the Range (image) and the Column Space.  Here for notes on matrix operations and here are notes relating matrices and transformations.  Similar material is in your text on pp. 204-246.
    17. 10/29/2018: You got back the second assignment, you can hand in a second draft of those to raise your grade but you need to get it to me no later than 11/8/2018.  We covered finding the matrix of a transformation of a matrix using the elementary basis \(\mathcal{E}\).  We also went over how to change between bases: \(\mathcal{B}\) to \(\mathcal{E}\), \(\mathcal{E}\) to \(\mathcal{B}\), and \(\mathcal{B}\) to \(\mathcal{D}\). Here are the notes on changing bases and here is the appropriate section in you text (p.254).
    18. 11/1/2018: Continued looking at change of bases and discussed orthonormal bases
    19. 11/5/2018: Continued discussion of orthogonal and orthonormal bases.  covered the Gram-Schmidt Orthogonalization process.  In you text the corresponding material is on pp. 267-277.  In the notes some material is here.
    20. 11/8/2018: Today we went over examples of using orthogonal projections to find least squares approximations.  We paid particular attention to how we think about it versus how we carry out the calculation.  See pp.277-292 in your text book and the notes on projections link to below.
    21. 11/12/2018: We introduced determinants, these are in chapter 4 of your text pay particular attention to pp.346-358 in your book.
    22. 11/15/2018: Practiced finding determinants and discussed some of their properties.
    23. 11/19/2018: Exam on Transformations and Matrix Operations and Stuff
    24. 11/26/2018: Similar Matrices and Diagonalizability: Online Notes and Text Reference (p.388)
    25. 11/29/2018: Eigenvalues and Eigenvectors: Text Reference (p.397)
    26. 12/3/2018: Wrap up: Look again at text reference (p.397) in your text, and here are some basic examples in the online notes like those in class.
    27. 12/6/2018: Determinants Exam – This is graded and you can come pick it up.
    28. 12/13/2018: Final Exam @ 11am in the usual place – **Study Guide and Practice Final Exam** – The practice final is optional and is worth extra credit.  Your final will have 15 questions, 5 vocabulary and 10 basic skills.
  • Assignments, Practice Exercises, and Practice Exams:
    1. Unit 1 on Solving Systems Due 9/24 for the practice exam & 9/27 for the assignment
    2. Unit 2 on Vector Spaces Due 10/15 for the practice exam & 10/18 for the assignment ←NEW DATES!!!
    3. Unit 3 on Matrices, Maps, and Projections Due 11/12 for the practice exam & 11/19 for the assignment & Exam on 11/19
    4. Unit 4 Practice Exam Information Due 12/6 and the exercises are due by the end of final exam week 12/14/2018 by noon.
  • Vocabulary:  Be sure to check out the Glossary included with the online notes.

    1. Unit 1: Matrix, Vector, Augmented Matrix, System of Equations, Vector Equation, Matrix Equation, Pivot Position, Consistent System, Inconsistent System, Homogeneous, Non-Homogeneous, Overdetermined, Underdetermined, Dot Product (Inner Product)
    2. Unit 2: Vector Space, Subspace, Linearly Independent, Linearly Dependent, Span of a set of Vectors, Column Space,  Row Space, Null space, Basis
    3. Unit 3: Linear Transformations, Matrix of a Transformation, One-to-One, Onto, Kernel, Zero Divisor, Transpose, Inverse Matrix, Null Space, Projection, Orthogonal Vectors, Orthogonal Complement, Projection of a vector onto a subspace, Orthonormal Basis, Change of Basis Matrix (also called change of coordinate matrix)
    4. Unit 4: Determinants, Cofactors, Classical Adjoint,  Eigenvector, Eigenvalue, Characteristic Polynomial, Similar Matrices, Diagonalizable
  • Extra Credit: You can earn up to +5% toward your final grade by completing the following …

    • Exercises (+0.5% for each problem):
      • Find examples of over determined systems with 0, 1, and infinitely many solutions.
      • Find examples of under determined systems with 0 and infinitely many solutions.  Argue that an under determined system can not have a unique solution.
      • Bring in flash cards for the unit vocabulary when you come in for the exam.
    • Note Taking (+1% for each day): You can earn up to 1% toward your final grade by volunteering to take class notes for a day. To be the note taker for the day you need to let me know ahead of time that you wish to do so, then after class you need to either type up your notes or write them up very neatly so that I can scan them and post them on the website for the rest of the class.
  • Links and Handouts:

The Power of Transformations: