 # MA211: Linear Algebra

Unit 2: Determinants, Rank, and Linear Transformations   *Despite the complicated definition of determinants, they are very useful because they allow you to determine, using only one number, whether or not a matrix is invertible.  They also are helpful for computing eigenvalues, which you will learn about later.  In this unit, you will learn about ways to find the determinants and to use determinants to compute the inverse of a matrix.

You will learn about the rank of the matrix, a very important concept in linear algebra.  You will begin by studying the row-reduced echelon form of a matrix and proving that the row-reduced echelon form for a given matrix is unique.  This is useful, because you can logically deduce important conclusions about the original matrix by examining its unique row-reduced echelon form.  You will then learn that the rank of a matrix is related to the number of linearly independent columns or rows of that matrix; it describes the dimensionality of the space.  It is also very important in the use of matrices to solve a system of linear equations, because it tells you whether Ax = 0 has zero, one, or an infinite number of solutions.

Finally, you will learn how matrices also arise in geometry, especially while studying certain types of linear transformations.  You will learn that a mxn matrix can be used to transform vectors in Fn to vectors in Fm via matrix multiplication.  As you will see, these types of transformation arise quite naturally in linear algebra and are important for applications in mathematics, physics, and engineering. *

Unit 2 Time Advisory
This unit should take you approximately 25 hours to complete.

***☐    Subunit 2.1: 7 hours***

***☐    Sub-subunit 2.1.1: 1 hour***

***☐    Sub-subunit 2.1.2: 6 hours***

***☐    Reading: 1 hour***

***☐    Assignment: 5 hours***

***☐    Subunit 2.2: 11 hours***

***☐    Sub-subunit 2.2.1: 1 hour***

***☐    Sub-subunit 2.2.2: 1 hour***

***☐    Sub-subunit 2.2.3: 1 hour***

***☐    Sub-subunit 2.2.4: 2 hours***

***☐    Sub-subunit 2.2.5: 6 hours***

***☐    Reading: 2 hours***

***☐    Assignment: 4 hours***

***☐    Subunit 2.3: 7 hours***

***☐    Reading: 3 hours***

***☐    Assignment: 4 hours***

Unit2 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- define and compute the ijth minor of a nxn matrix; - define and compute the ijth cofactor of a nxn matrix; - find the determinant by expanding along rows and columns; - use the method of Laplace expansion; - state properties of determinants; - find determinants using row operations; - find the inverse of a matrix; - state and use Cramer’s rule to find a solution to a system of equations; - define elementary matrices; - define rank (determinant rank, row rank, column rank); - find the rank of a matrix; - obtain the row-reduced echelon form for a matrix; - define and verify linear independence; - determine whether a given set of vectors is linearly independent; - define span, basis, and dimension for a vector space; - extend an independent set of vectors to form a basis; - define and find the kernel of a matrix; - define, compute, and find the null space of a matrix; - state and apply the rank-nullity theorem; - identify the relation between rank and existence of solutions to linear systems; - determine whether a given set is a subspace; - compute the dimension of a space; - define and compute linear transformations; - construct the matrix of a linear transformation; - construct the matrix for a projection; - construct the matrix for a rotation; - find the general solution to a linear system; - discuss row equivalence and use row and column space to solve linear systems; - define spanning set and determine the span of a set of vectors; - show that a set of vectors is a basis; - define and compute column space, row space, nullspace, and rank; - describe how the determinant changes as a result of elementary row operations; - compute the determinant using cofactor expansions; - compute the determinant row reduction; and - compute the determinant using Cramer’s rule.

2.1 Determinants   2.1.1 Basic Techniques and Properties   - Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.1: Basic Techniques and Properties” Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.1: Basic Techniques and Properties” (PDF)

Instructions: Please click on the link above, and read Section 6.1 on pages 97–104.  Section 6.1 will provide an introduction to determinants and techniques for finding them.  This reading should take you approximately 1 hour to complete.

2.1.2 Applications   - Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.2: Applications” Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.2: Applications” (PDF)

Instructions: Please click on the link above, and read Section 6.2 on pages 104–109.  Section 6.2 will introduce some applications, including Cramer’s rule.  This reading should take you approximately 1 hour to complete.

• Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.3: Exercises” Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.3: Exercises” (PDF)

Instructions: Please click on the link above to open the PDF.  Scroll down to page 109, and complete problems 1, 3, 4, 5, 9, 11, 13, 16, 20, 21, 24, 26, 28, 32, and 36.  Next, click on “Solutions” (PDF) and check your answers on pages 28–38.  This assessment should take you approximately 5 hours to complete.

2.2 Rank of a Matrix   2.2.1 Elementary Matrices   - Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.1: Elementary Matrices” Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.1: Elementary Matrices” (PDF)

Instructions: Please click on the link above, and read Section 8.1 on pages 129–134.  Section 8.1 will introduce the elementary matrices, which result from doing a row operation to the identity matrix.  This reading should take you approximately 1 hour to complete.

2.2.2 The Row Reduced Echelon Form of a Matrix   - Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.2: The Row Reduced Echelon Form of a Matrix” Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.2: The Row Reduced Echelon Form of a Matrix” (PDF)

Instructions: Please click on the link above, and read Section 8.2 on pages 135–139.  Section 8.2 will review the description of the row-reduced echelon form.  This reading should take you approximately 1 hour to complete.

2.2.3 The Rank of a Matrix   - Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.3: The Rank of a Matrix” Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.3: The Rank of a Matrix” (PDF)

Instructions: Please click on the link above, and read Section 8.3 on pages 139–142.  Section 8.3 will define rank and explain how to find the rank.  This reading should take you approximately 1 hour to complete.

2.2.4 Linear Independence and Bases   - Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.4: Linear Independence and Bases” Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.4: Linear Independence and Bases” (PDF)

Instructions: Please click on the link above, and read Section 8.4 on pages 142–152.  Section 8.4 will introduce linear independence and bases.  This reading should take you approximately 2 hours to complete.

2.2.5 Fredholm Alternative   - Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.5: Fredholm Alternative” Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.5: Fredholm Alternative” (PDF)

Instructions: Please click on the link above, and read Section 8.5 on pages 153–156.  Section 8.5will introduce the Fredholm Alternative for the case of real matrices here.  This reading should take you approximately 2 hours to complete.

• Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.6: Exercises” Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.6: Exercises” (PDF)

Instructions: Please click on the link above to open the PDF.  Scroll down to page 156, and complete problems 2, 5, 7, 10, 12, 16, 18, 25, 32, 34, 45, 50, and 54.  Next, click on “Solutions” (PDF) and check your answers on pages 38–49.  This assessment should take you approximately 4 hours to complete.