 # MA212: Linear Algebra II

Unit 1: The Basics of Linear Algebra   This unit serves as a review of some of the material covered in Linear Algebra I, including linear equations, matrices, and determinants.  Specifically, you will review properties of the real numbers and complex numbers.  You will then learn the Fundamental Theorem of Algebra, which states that every polynomial equation in one variable with complex coefficients has at least one complex solution.  You will also review how to solve linear systems of equations and perform operations on matrices.  The key is to read through all the material below and complete all the exercises in this unit.  The goal of this unit is to ensure that you are comfortable with the key matrix algebra concepts related to Euclidean spaces as these concepts will be referred to throughout this course.   The skills and techniques you learn working with matrix theory will be generalized later in the course when you work in a more abstract linear algebra setting.

Unit 1 Time Advisory
This unit will take you approximately 32.5 hours to complete.

☐    Subunit 1.1: 11.5 hours ☐    Sub-subunit 1.1.1: 1 hour

☐    Sub-subunit 1.1.2: 1 hour

☐    Sub-subunit 1.1.3: 5 hours

☐    Sub-subunit 1.1.4: 2.5 hours

☐    Sub-subunit 1.1.5: 1 hour

☐    Sub-subunit 1.1.6: 1 hour

☐    Subunit 1.2: 3.5 hours

☐    Subunit 1.3: 11 hours ☐    Sub-subunit 1.3.1: 5.5 hours

☐    Sub-subunit 1.3.2: 5.5 hours

☐    Subunit 1.4: 6.5 hours ☐    Reading: 1.5 hours

☐    Activity: 5 hours

Unit1 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Solve systems of equations using matrix methods. - Perform operations on matrices. - Identify properties of matrix multiplication. - State the definition of ordered fields. - State which properties are necessary for the complex numbers to form a field. - State the Archimedean property. - Identify an example of a space that has the Archimedean property - Define the inner product, and use it to compute lengths and angles. - State and apply the triangle inequality. - Use the polar form and geometric interpretation of the complex numbers to solve problems. - Explain what the fundamental theorem of algebra states. - Explain why the complex numbers are important in the context of fundamental theorem of algebra.

1.1 Preliminaries   1.1.1 Sets and Functions   - Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Appendix A: The Language of Sets and Functions”

``````Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne
Schilling’s *Linear Algebra: As an Introduction to Abstract
Mathematics*: [“Appendix A: The Language of Sets and
(PDF)

Instructions: Please click on the link above, select the “PDF
version of book” link, and read Appendix A for a definition of sets
and functions as well as to learn about associated vocabulary for
these concepts.  You will be using this text throughout the course,
so it may help to save the PDF to your desktop for easy reference.

Studying this reading should take approximately 1 hour to
complete.

displayed on the webpage above.
``````

1.1.2 The Number Line, Real Numbers and Ordered Fields   - Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries”

``````Link: Brigham Young University: Kenneth Kuttler’s *An Introduction
to Linear Algebra:* [“Chapter 1:
(PDF)

Instructions: Please read Sections 1.1–1.4 in their entirety (pages
11–15).  This reading should be a review.

Studying this reading should take approximately 1 hour to
complete.

Terms of Use: *An Introduction to Linear Algebra* was written by
Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the
Saylor Foundation’s Open Textbook Challenge.
``````

1.1.3 Complex Numbers   - Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries”

``````Link: Brigham Young University: Kenneth Kuttler’s *An Introduction
to Linear Algebra:* [“Chapter 1:
(PDF)

Instructions: Please read Section 1.5 in its entirety (pages
15–19).  Pay particular attention to the polar decomposition of
complex numbers and the associated geometric interpretation.

Studying this reading should take approximately 30 minutes to
complete.

Terms of Use: *An Introduction to Linear Algebra* was written by
Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the
Saylor Foundation’s Open Textbook Challenge.
``````
• Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 2: Introduction to the Complex Numbers”

Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 2: Introduction to the Complex Numbers” (PDF)

Instructions: If you have not already downloaded the text, please click on the link above.  Read Chapter 2.  Pay particular attention to the polar decomposition of complex numbers and the associated geometric interpretation.

Studying this reading should take approximately 1 hour to complete.

Terms of Use: These materials have been reproduced for educational and non-commercial purposes, and can be viewed in their original format here.  Any reproduction or redistribution for commercial use is strictly prohibited.

• Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 2”

Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 2” (PDF)

Instructions: If you have not already downloaded and saved the text, please click on the link above. Complete calculational exercises 1(a, b, c), 2(b, d), and 4(a, b) and the proof-writing exercises 1, 2, 3, 4, 5 (pages 24 and 25).

Completing this activity should take approximately 3.5 hours to complete.

Terms of Use: These materials have been reproduced for educational and non-commercial purposes, and can be viewed in their original format here.  Any reproduction or redistribution for commercial use is strictly prohibited.

1.1.4 The Fundamental Theorem of Algebra   - Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “The Fundamental Theorem of Algebra and Factoring Polynomials”

``````Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne
Schilling’s *Linear Algebra: As an Introduction to Abstract
Mathematics*: [“The Fundamental Theorem of Algebra and Factoring
(PDF)

Instructions: If you have not already downloaded and saved the
text, please click on the link above. Read Chapter 3. Here, you will
read about the important Fundamental Theorem of Algebra.  In
particular, note how the theorem is false when considering
polynomials in the real number system.

Studying this reading should take approximately 1 hour to
complete.

Terms of Use: These materials have been reproduced for educational
and non-commercial purposes, and can be viewed in their original
format
[here](https://mail.whittier.edu/owa/redir.aspx?C=fa77263df66842f9a74d8d1b871cd15b&URL=http%3A%2F%2Fwww.math.ucdavis.edu%2F~anne%2Flinear_algebra%2Fmat67_course_notes.pdf).
Any reproduction or redistribution for commercial use is strictly
prohibited.
``````
• Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 3”

Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 3” (PDF)

Instructions: If you have not already downloaded and saved the document, please click on the link above. Complete calculational exercise 2 and the proof-writing exercise 3 (pages 34 and 35).

Completing this activity should take approximately 1.5 hours to complete.

Terms of Use: These materials have been reproduced for educational and non-commercial purposes, and can be viewed in their original format here.  Any reproduction or redistribution for commercial use is strictly prohibited.

1.1.5 Properties of the Real Numbers   - Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries”

``````Link: Brigham Young University: Kenneth Kuttler’s *An Introduction
to Linear Algebra:* [“Chapter 1:
(PDF)

Instructions: Please read Sections 1.7–1.9 (pages 20–26) in their
entirety. Here, you will learn some analytic properties that
uniquely characterize the real number system.

Studying this reading should take approximately 1 hour to complete.

Terms of Use: *An Introduction to Linear Algebra* was written by
Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the
Saylor Foundation’s Open Textbook Challenge.
``````

1.1.6 Systems of Equations   - Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries”

``````Link: Brigham Young University: Kenneth Kuttler’s *An Introduction
to Linear Algebra:* [“Chapter 1:
(PDF)

Instructions: Please read Sections 1.10–1.13 (pages 26–33) in their
entirety.  Most of this material should be a review.

Studying this reading should take approximately 1 hour to
complete.

Terms of Use: *An Introduction to Linear Algebra* was written by
Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the
Saylor Foundation’s Open Textbook Challenge.
``````

1.2 What Is Linear Algebra?   - Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “What Is Linear Algebra?”

``````Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne
Schilling's *Linear Algebra: As an Introduction to Abstract
Mathematics*: [“What Is Linear
(PDF)

Instructions: Please click on the link above, and then read Section
1.2 titled “What Is Linear Algebra?”

Studying this reading should take approximately 30 minutes to
complete.

Terms of Use: These materials have been reproduced for educational
and non-commercial purposes and can be viewed in their original
format
[here](https://mail.whittier.edu/owa/redir.aspx?C=fa77263df66842f9a74d8d1b871cd15b&URL=http%3A%2F%2Fwww.math.ucdavis.edu%2F~anne%2Flinear_algebra%2Fmat67_course_notes.pdf).
Any reproduction or redistribution for commercial use is strictly
prohibited.
``````
• Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 1”

Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 1” (PDF)

Instructions: If you have not already downloaded and saved the document to your desktop, please click on the link above and select the “PDF version of book” link.  Complete exercises 1b and 2 on page 9 and proof-writing exercise 1 on page 10.

Completing this activity should take approximately 3 hours to complete.

Terms of Use:  These materials have been reproduced for educational and non-commercial purposes and can be viewed in their original format here.  Any reproduction or redistribution for commercial use is strictly prohibited.

1.3 Matrices and Linear Transformations   1.3.1 Matrices   - Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Supplementary Notes on Matrices and Linear Systems”

``````Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne
Schilling’s *Linear Algebra: As an Introduction to Abstract
Mathematics*: [“Supplementary Notes on Matrices and Linear
(PDF)

Instructions: If you have not already downloaded and saved the
Sections 12.1–12.3 in their entirety.  Most of this material should
be a review.

Studying this reading should take approximately 1 hour to
complete.

Terms of Use: These materials have been reproduced for educational
and non-commercial purposes, and can be viewed in their original
format
[here](https://mail.whittier.edu/owa/redir.aspx?C=fa77263df66842f9a74d8d1b871cd15b&URL=http%3A%2F%2Fwww.math.ucdavis.edu%2F~anne%2Flinear_algebra%2Fmat67_course_notes.pdf).
Any reproduction or redistribution for commercial use is strictly
prohibited.
``````
• Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations”

Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations” (PDF)

Instructions: Please read Section 2.1 (pages 37–51) in its entirety.  Here, you will review some basic properties of matrix algebra.

Studying this reading should take approximately 30 minutes to complete.

Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.

• Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations”

Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations” (PDF)

Instructions: Please work through the odd-numbered problems for 11–29 in Section 2.2 of the textbook on pages 52 and 53. When you are done, check your solutions with the answers on page 487.

Completing this activity should take approximately 4 hours to complete.

Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.

1.3.2 Maps and Spaces   - Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations”

``````Link: Brigham Young University: Kenneth Kuttler’s *An Introduction
to Linear Algebra:* [“Chapter 2: Matrices and Linear
(PDF)

Instructions: Please read Sections 2.3–2.6 (pages 53–71) in their
entirety.  Pay particular attention to the relationship between
linear transformations and matrices.

Studying this reading should take approximately 45 minutes to
complete.

Terms of Use: *An Introduction to Linear Algebra* was written by
Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the
Saylor Foundation’s Open Textbook Challenge.
``````
• Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations”

Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations” (PDF)

Instructions: Please work through the odd-numbered problems for 1–25 in Section 2.7 on pages 71–73.  When you are done, check your solutions with the answers on page 488.

Completing this activity should take approximately 4 hours to complete.

Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.

• Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Supplementary Notes on Matrices and Linear Systems”

Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Supplementary Notes on Matrices and Linear Systems” (PDF)

Instructions: If you have not already downloaded and saved the PDF document, click on the link above and select the “PDF version of book” link.  Read Sections 12.4 and 12.5 in their entirety.  Note the relationship between matrices and linear transformations.

Studying this reading should take approximately 45 minutes to complete.

Terms of Use: These materials have been reproduced for educational and non-commercial purposes, and can be viewed in their original format here.  Any reproduction or redistribution for commercial use is strictly prohibited.

1.4 Vectors and Matrices   - Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 3: Determinants”

``````Link: Brigham Young University: Kenneth Kuttler’s *An Introduction
to Linear Algebra:* [“Chapter 3:
(PDF)

Instructions: Please read Chapter 3 (pages 77–104) in its entirety.
The determinant of a matrix is an extremely important number
associated to the matrix as it provides us with a lot of information
about the associated linear transformation.

Studying this reading should take approximately 1.5 hours to
complete.

Terms of Use: *An Introduction to Linear Algebra* was written by
Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the
Saylor Foundation’s Open Textbook Challenge.
``````
• Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 3: Determinants”

Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 3: Determinants” (PDF)

Instructions: Please click on the link above, and work through problems 2, 3, 6, 9, 11, 13, and 15 in Section 3.2 (pages 82 and 83) and problems 5, 6, 8, 9, and 11 in Section 3.6 (pages 102 and 103). When you are done, check your solutions with the answers on page

1.
Completing this activity should take approximately 5 hours.

Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.