Unit 1: Vector Products, Systems of Equations, and Matrices
*This unit begins with a review of vectors. You will learn the geometric meaning of vectors, which is especially significant in R^{2} and R^{3}. Next, you will learn the geometric meaning of vector addition and scalar multiplication. Finally, you will apply study vectors in the context of physics to model force and other physical vectors like velocity.
In the next chapter, you will begin learning about vector products. There are two ways of multiplying vectors, both of which are of great importance in applications. The first type of a product is called the dot product, also called the scalar product or the inner product. You will then study the geometric significance of the dot product and applications of dot product by studying the concepts of work and projections. Next, you will begin the study of cross products. The cross product is the other way of multiplying vectors, and it is different from the dot product in fundamental ways. You will learn both the geometric meaning of the cross product and the description in terms of coordinates. Both descriptions of the cross product are important; the geometric description is necessary to understand the applications to physics and geometry while the coordinate description is necessary to actually compute the cross product. You will then learn techniques, which will allow you to discover vector identities and simplify expressions involving cross and dot products in three dimensions.
Next, you will begin exploring systems of linear equations. The basic idea is to study situations where there are several different variables that are related in multiple ways. These linear equations could describe budget constraints in a business, physical constraints in an engineering problem, or any number of other situations. The key is that these constraints can be described by linear equations. The geometric interpretation of these constraints is that each equation describes a line or plane where potential solutions to the problem must lie. The task then is to figure out what combination of variable values solves all of the different linear equations at the same time. Geometrically, this is where all of the lines or planes intersect. Just as is the case with the problems that the equations may be modeling, the system of equations will sometimes have no solution, will sometimes have a single solution, and will sometimes have an infinite number of solutions. Finally, you will learn about matrices andhow to write a system of linear equations as a matrix equation. While this may have at first appeared to be merely a way of putting your coefficients in a table, matrices in fact have many interesting (but not immediately obvious!) properties.*
Unit 1 Time Advisory
This unit should take you approximately 28.5 hours to complete.
☐ Subunit 1.1: 6 hours
☐ Sub-subunit 1.1.1: 1 hour
☐ Sub-subunit 1.1.2: 2 hours
☐ Sub-subunit 1.1.3: 3 hours
☐ Subunit 1.2: 9 hours
☐ Sub-subunit 1.2.1: 4 hours
☐ Reading: 1.5 hours
☐ Assignment: 2.5 hours
☐ Sub-subunit 1.2.2: 1 hour
☐ Sub-subunit 1.2.3: 4 hours
☐ Reading: 0.5 hour
☐ Assignment: 3.5 hours
☐ Subunit 1.3: 5.5 hours
☐ Sub-subunit 1.3.1: 0.5 hour
☐ Sub-subunit 1.3.2: 5 hours
☐ Reading: 1.5 hours
☐ Assignment: 3.5 hours
☐ Subunit 1.4: 8 hours
☐ Reading: 3 hours
☐ Assignment: 5 hours
Unit1 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- explain and perform algebraic operations done with elements in
F^{n};
- explain the geometric meaning of vectors;
- explain the geometric meaning of vector addition;
- compute the distance between points in R^{n};
- compute the length of a vector;
- explain the geometric meaning of scalar multiplication;
- draw a picture of points in R^{2} and R^{3} given
ordered pairs;
- explain how vectors are used to define force and velocity;
- prove the Cauchy-Schwarz inequality;
- explain the geometric significance of the dot product;
- explain the geometric significance of the cross product;
- define and compute work;
- find the projection of a vector v onto a vector u;
- state properties of the inner product;
- find the angle between two vectors;
- define and compute the cross product, the dot product, and the box
product;
- explain the distributive law for the cross product;
- compute the volume of the parallelepiped;
- define the Kronecker delta;
- find solutions to a system of linear equations, both algebraically
and graphically; and
- use the Gauss elimination.
1.1 Fn and Vector Products
1.1.1 Vectors and their Geometric Meaning
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 2: Fn:” “Section 2.1: Algebra in Fn,” “Section 2.2:
Geometric Meaning of Vectors,” and “Section 2.3: Geometric Meaning
of Vector Addition”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
2: Fn:” “Section 2.1: Algebra in Fn.” “Section 2.2: Geometric
Meaning of Vectors,” and “Section 2.3: Geometric Meaning of Vector
Addition” (PDF)
Instructions: Please click on the link above, and read the
indicated sections on pages 23–27. Section 2.1 introduces you to
algebraic operations done with elements of Fn. In Section 2.2,
you will explore the geometric meaning of vectors, and in Section
2.3, you will study the geometric interpretation of vector
addition. These readings should take you approximately 1 hour to
complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
1.1.2 Length of a Vector and Scalar Multiplication
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 2: Fn:” “Section 2.4: Distance between Points in Length of
a Vector” and “Section 2.5: Geometric Meaning of Scalar
Multiplication”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
2: Fn.” “Section 2.4: Distance between Points In Rn Length of a
Vector” and “Section 2.5: Geometric Meaning of Scalar
Multiplication” (PDF)
Instructions: Please click on the link above, and read the
indicated sections on pages 27–31. In Section 2.4, you will study
how distance is defined between two points in Rn. In Section 2.5,
you will explore the geometric meaning of scalar multiplication.
These readings should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra:
"Chapter 2:" “Section 2.6: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
2: Fn.” “Section 2.6: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 31 to Section 2.6, and complete problems 1, 2, 4, and 5. Next, click on “Solutions” (PDF) and check your answers on pages 5–6. This assessment should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
1.1.3 Vectors and Physics - Reading: Professor Kenneth Kuttler's Elementary Linear Algebra: "Chapter 2: Fn." "Section 2.7: Vectors and Physics"
Link: Professor Kenneth Kuttler’s *[Elementary Linear
Algebra:](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/04/Elementary-Linear-Algebra-4-26-12-Kuttler-OTC.pdf) *“Chapter
2:” “Section 2.7: Vectors and Physics” (PDF)
Instructions: Please click on the link above, and read Section 2.7
on pages 32–36. In this section, you will learn about the concept
of force. This reading should take you approximately 1 hour to
complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
Assessment: Professor Kenneth Kuttler's Elementary Linear Algebra: "Chapter 2: Fn." "Section 2.8: Exercises" Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 2: Fn." “Section 2.8: Exercises”(PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 36, and complete problems 2, 3, 7, 9, 10, and 11. Next, click on “Solutions” (PDF) and check your answers on pages 6–8. This assessment should take you approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
1.2 Vector Products
1.2.1 The Dot Product
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 3: Vector Products:” “Section 3.1: The Dot Product” and
“Section 3.2: The Geometric Significance of the Dot Product”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
3: Vector Products:” “Section 3.1: The Dot Product” and “Section
3.2: The Geometric Significance of the Dot Product” (PDF)
Instructions: Please click on the link above, and read the
indicated sections on pages 39–47. Section 3.1 will provide the
definition and properties of the dot product. Section 3.2 will
discuss the geometric meaning of the dot product and then apply the
ideas to the concepts of work and projection. These readings should
take you approximately 1 hour and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 3: Vector Products:” “Section 3.3: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
3: Vector Products:” “Section 3.3: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 47, and work on problems 1, 2, 4, 6, 10, 14, 15, 17, 20, and 21. Next, click on “Solutions” (PDF) and check your answers on pages 8–10. This assessment should take you approximately 2 hours and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
1.2.2 The Cross Product
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 3: Vector Products:” “Section 3.4: The Cross Product”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
3: Vector Product:” “Section 3.4: The Cross Product” (PDF)
Instructions: Please click on the link above, and read Section 3.4
on pages 48–54. Section 3.4 will provide the definition,
properties, and the geometric meaning of the cross product. This
reading should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
1.2.3 The Vector Identity Machine
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 3: Vector Products:” “Section 3.5: The Vector Identity
Machine”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
3: Vector Products:” “Section 3.5: The Vector Identity Machine”
(PDF)
Instructions: Please click on the link above, and read Section 3.5
on pages 54–56. Section 3.5 will introduce a technique that will
allow you to discover vector identities and simplify expressions
involving cross and dot products in three dimensions. This reading
should take you approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 3: Vector Products:” “Section 3.6: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
3: Vector Products:” “Section 3.6: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 56, and complete problems 1, 4, 6, 7, 8, 9, 10, 13, 16, 18, and 20. Next, click on “Solutions” (PDF) and check your answers on pages 11–14. This assessment should take you approximately 3 hours and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
1.3 Systems of Equations
1.3.1 Systems of Equations, Geometry
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 4: Systems of Equations:” “Section 4.1: Systems of
Equations, Geometry”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
4: Systems of Equations:” “Section 4.1: Systems of Equations,
Geometry” (PDF)
Instructions: Please click on the link above, and read Section 4.1
on pages 59–61. Section 4.1 will explore how to find solution(s)
for a system of linear equations by graphing. This reading should
take you approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
1.3.2 Systems of Equations, Algebraic Procedures
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 4: Systems of Equations:” “Section 4.2: Systems of
Equations, Algebraic Procedures”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: Chapter
4: Systems of Equations:” “Section 4.2: Systems of Equations,
Algebraic Procedures” (PDF)
Instructions: Please click on the link above, and read Section 4.2
on pages 61–72. Section 4.2 will explore how to find solution(s)
for a system of linear equations using elementary operations,
Gaussian elimination, and other algebraic procedures. This reading
should take you approximately 1 hour and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 4: Systems of Equations:” “Section 4.3: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
4: Systems of Equations:” “Section 4.3: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 72, and complete problems 1, 6, 12, 13, 17, 19, 22, 30, 35, and 36. Next, click on “Solutions” (PDF) and check your answers on pages 14–20. This assessment should take you approximately 3 hours and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
1.4 Matrices
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 5: Matrices:” “Section 5.1: Matrix Arithmetic”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
5: Matrices:” “Section 5.1: Matrix Arithmetic” (PDF)
Instructions: Please click on the link above, and read Section 5.1
on pages 77–91. Section 5.1 will provide an overview of matrices
and matrix arithmetic. This reading should take you approximately 3
hours to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 5: Matrices:” “Section 5.2: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
5: Matrices:” “Section 5.2: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 92, and complete problems 3, 5, 6, 7, 10, 12, 16, 19, 24, 28, 32 37, 40, 44, 47, and 53. Next, click on “Solutions” (PDF) and check your answers on pages 19–28. This assessment should take you approximately 5 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.