MATH-2318 Linear Algebra and Matrix Theory

Gustavo Cepparo

Credit Spring 2024

Section(s)

MATH-2318-001 (76768)
LEC MW 6:00pm - 7:20pm DIL DLS DIL

Course Requirements

Homework: : Assigned daily and collected the following week on Monday and Thursdays. I will not grade any disorganized or difficult-to-read assignments. Your homework is your best piece of work—so do it every day. I will not accept homework in loose sheets of paper. The lowest homework score will be dropped. No late homework will be accepted.

Students will have the option of replacing the lowest test score on the first three tests with the final (test 4).

All students must take test 4.

Grading: Homework 15 %

Tests 20 % each

Final exam 25 %

Grading. A: 90-100; B: 80-89; C: 70-79; D: 60-69; F: below 60

Making Time to Learn

We learn math by thinking about and working on mathematical problems, which takes time. Practice is crucial in a math course. To ensure that you have adequate time, set aside 8-12 hours per week outside of class time to practice and study for this course. Ask for help immediately when something isn’t clear.

Getting Help

ACC provides several free resources for students who need help; descriptions and links are below:

Office hours: Another name for office hours is “student hours.” This is the time your instructor has set aside to answer student questions, so feel free to drop by if you have questions. Office hours may be virtual or on campus; see information above.

Instructional Associates: Instructional Associates specific to the course you are taking are available for tutoring. To make an appointment, go to https://sites.google.com/a/austincc.edu/math-students/meet/list and then click on your course.

Learning Labs: The ACC Learning Labs provide tutoring in math and other subjects. To schedule an appointment, go to https://www.austincc.edu/students/learning-lab. This site includes information about in person and virtual tutoring options.

Academic Coaching: Academic coaches offer extra support to students with study strategies; they want to help you learn to be an active participant in your own learning process. For more information or to make an appointment with an academic coach, go to https://www.austincc.edu/students/academic-coaching.

ACC Student Services: Services are offered in many areas, including Academic, Financial, Personal, and Technology Support. For more information, go to https://www.austincc.edu/student-support.

Readings

REQUIRED TEXTS/MATERIALS

The required textbook for this course is:

Text:Linear Algebra, A Geometric Approach, 2nd edition, by Theodore Shifrin & Malcolm R. Adams, W. H. Freeman & Co, 2011. ISBN-10: 1-4292-1521-6, ISBN-13: 978-1-4292-1521-3, Cloth Text, 464 pages.

Computers and calculators

The use of calculators or computers in order to perform routine computations is encouraged in order to give students more time on abstract concepts.

Course Subjects

Linear Algebra Calendar with Homework– Math 2318

Note: Schedule changes may occur during the semester. Any changes will be announced in class and posted as a Blackboard Announcement

Week 1

Lecture 1 (Jan 17)

Introduction and Syllabus

Write definitions in a flash card for easy access during homework, lecture, and office hours.

Section 1.2 Dot Product

[1] Definition (page 19)

[2] Proposition 2.1

[3] Corollary 2.2

Homework 1 (four problems)

Section 1.2

Problems:

[1] 10 (page 26)

[2] 11

[3] 13

[4] 18

Week 2

Lecture 2 (Jan 22)

[1] A Conditional Statement (Logic) (page 21)

[2] Definition (page 20)

[3] The Construction! (page 22)

[4] Proposition 2.3 (Cauchy-Schwarz)

Homework 2

Section 1.1

Problems:

[1] 8 (page 15)

[2] 20

[7] 22

[3] 23

[4] 24

[5] Show the argument below is valid (c is a contradiction).

Lecture 3 (Jan 24)

Section 1.1

Write definitions in a card for easy access during homework, lecture, and office hours.

[1] Definitions (page 2 and 3)

[2] Lines (page 7)

[3] Example 3 and 4

[4] Example 5

Week 3

Lecture 4 (Jan 29)

Section 1.1

[1] Definition (page 11)

[2] Definition (page 12)

[3] Examples 6 to 11

[4] The Row View, Column View and Matrix View.

Homework 3 (three problems)

Section 1.3

Problems:

[1] 6 (page 35)

[2] 8

[3] 12 (a) and (c)

Lecture 5 (Jan 31)

Section 1.3 Hyperplanes

[1] Example 1 (page 29)

[2] Distance from plane to origin.

[3] Example 2 and 3 (Cartesian to Parametric.

[4] Get the intersection between two planes in R^3: A plane or line.

[4] Example 4

[5] Example 5

Week 4

Lecture 6 (Feb 5)

Section 1.4 Systems of Linear Equations and Gausian Elimination

[1] Example 1

[2] Example 2

[3] Definition (page 49)

[4] Example 5

Homework 4

Section 1.4

Problems:

[1] 3 (a), (b) and (d) (page 50)

[2] 10

[3] 14

[4] 15

Lecture 7 (Feb 7)

Review for Test 1

Week 5

Lecture 8 (Feb 12)

Test 1

TEST 1 Monday, February 12th Note: For Test 1 you will need a webcam a phone with a camera and Zoom.

Lecture 9 (Feb 14)

Section 1.5 The Theory of Linear Systems

[1] Example 1

[2] Example 2

[3] Definition (page 55)

[4] Constraint Equations (From Parametric to Cartesian)

Week 6

Lecture 10 (Feb 19)

Section 1.5 The Theory of Linear Systems

[1] Example 3

[2] Example 4

[3] Example 5

[4] Uniqueness and Non-uniqueness of Solutions

[5] Definition (page 59)

[6] Theorem 5.3

[7] Definition (page 61) Singular and Non-singular

Homework 5 (six problems)

Section 1.5

Problems:

[1] 3 a (page 62)

[2] 4 a and c

[3] 8

[4] 10

[5] 13

[6] 15

Lecture 11 (Feb 21)

Section 1.6 Applications

[1] No-int Model

[2] Stoichiometry

Week 7

Lecture 12 (Feb 26)

Section 2.1

[1] Four ways to multiply matrices

[2] A factorization A = C R

Homework Week 6

Section 1.6 and 2.1

Problems:

Section 1.6

[1] 12 a

[2] Multiply 4 different ways (AB):

[9] Factor the matrix B above into C times R. That is B = C R

Lecture 13 (Feb 28)

Section 2.2

[1] Linear Transformations

[2] Example 1 (A projection)

[3] Example 2 (A rotation)

Week 8

Lecture 14 (Mar 4)

[1] Section 2.2 continue Linear Transformations

Section 2.3 Inverse Matrices

[2] Example 2

[3] Example 3

[4] Proposition 3.4

Homework Week 7 (six problems

Section 2.2

Problems:

[1] 4 c and f (page 101)

[2] 5 b

[3] 10

Section 2.3

Problems:

[4] 2 b (page 108)

[5] 4

[6] 5

Lecture 15 (Mar 6)

Section 2.4 Elementary Matrices and the A = LU factorization

[1] Example 1

[2] Example 5

Spring Break Holiday, Monday, March 11th – Sunday, March 17th

Week 9

Lecture 16 (Mar 18)

Section 2.5 The Transpose

[1] Proposition 5.1

[2] Proposition 5.2

Homework 8

Section 2.4

Problems:

[7] 4 (page 118)

[8] 11 (page 122)

[9] 15 (page 122)

[10] 1 a, b, d, g (page 134)

Lecture 17 (Mar 20)

Section 3.1 Subspaces of R^n

[1] Definition (page 127)

[2] Proposition 1.1

[3] Proposition 1.2

[4] Example 3

[5] Example 5

[6] Definition (page 133)

[7] Proposition 1.3

[8] Example 6

[9] Definition (page 134)

Week 10

Lecture 18 (Mar 25)

Section 3.2 The Four Fundamental Subspaces of a Matrix A

[1] Four Definitions

[2] Proposition 2.1 (page 136)

[3] Proposition 2.2 (page 139)

[4] Example 1

Homework 9

Section 3.1

Problem:

[1] 4 (page 135)

[2] 6

[3] 11

Lecture 19 (Mar 27)

No class

Week 11

Lecture 20 (April 1)

TEST 2 Wednesday, March 27th

Note: For Test 2 you will need a webcam a phone with a camera and Zoom.

Lecture 21 (April 3)

Linear Independence and Basis

Section 3.3

Read Definitions “linearly independent set of vectors”, “basis”, and “coordinates of an ordered basis”

[1] Example 1 (pg 143)

[2] Proposition 3.1

[3] Example 2 (pg 146)

[4] Example 3 and 4

[5] Proposition 3.2

and continue

Linear Independence and Basis

Section 3.3

[1] Proposition 3.2 (pg 149)

[2] Definition (pg 149)

[3] Example 7

[4] Corollary 3.3

[5] Definition (pg 150)

[6] Example 8

[7] Proposition 3.4

[8] Example 9

Homework 10 (four problems)

Section 3.2

Problems:

[1] Find the four fundamental spaces of

[2] Prove proposition 2.3 (page 149)

[3] 9 (page 143)

[4] 13 a

Week 12

Lecture 22 (April 8)

Dimension and Its Consequences

Section 3.4

Read definitions: “Dimension”

[1] Proposition 4.1

[2] Example 1

[3] Theorem 4.2

[4] Proposition 4.3

[5] Example 2

[6] Theorem 4.6

[7] Corollary 4.7 (Nullity-Rank Theorem)

[8] Theorem 4.9

[9] Example 5

Homework 11 (four problems)

Section 3.3

Problems:

[1] Problem 1

[2] Problem 2c

[3] Problem 3

[4] Problem 8

Lecture 23 (April 10)

Abstract Vector Spaces

Section 3.6

[1] Definition (pg 176)

[2] More Important Definition (pg 177)

[3] Example 1

[4] Proposition 6.1

[5] Example 2

[6] Example 4

[7] Example 6

[8] Proposition 6.3

[9] Example 8 (continue from Prop. 6.3)

Week 13

Lecture 24 (April 15)

Abstract Vector Spaces

Section 3.6

[1] Example 9

[2] Definition of an Inner Product Space

[3] Example 10

[4] Example 11

Homework 12

Section 3.4

Problems:

[1] Problem 1a

[2] Problem 1b

[3] Problem 11a

[4] Problem 8b

Section 3.6

Problems:

[5] Problem 2a

[6] Problem 2c

[7] Problem 3a

[8] Problem 6b

Lecture 25 (April 17)

Inconsistent Systems and Projection

Section 4.1

[1] Definition (pg 192)

[2] Definition (pg 193)

[3] Example 1

[4] Example 2

[5] Example 3

[6] Example 4

Data Fitting (pg 196) and Orthogonal Bases (pg 200)

Section 4.2

[1] Example 5 Data Fitting (Regression)

[2] Example 1 (pg 203) Look at Example 4 from Monday, April 20 (again)

[3] Gram-Schmidt Process (pg 204)

[4] The QR Decomposition

Week 14

Lecture 26 (April 22)

TEST 3 Monday, April 26th Note: For Test 3 you will need a webcam a phone with a camera and Zoom.

Homework 13

Section 4.1

Problems:

[1] Problem 1a

[2] Problem 2

[3] Problem 14

Data Fitting and Section 4.2

Problems:

[4] Problem 2a

[5] Problem 8a

[6] Problem 10

Section 5.1

Problems:

[7] Use the 2 by 2 matrix from Example 3 on page 250 in order to calculate the determinant using the three properties.

Section 5.2

Problems:

[8] Problem 1 (pg 252)

Lecture 27 (April 24)

Linear Transformations on Abstract Vector Spaces

Section 4.4

[1] Example 1

[2] Definition on page 225

[3] Example 5

Properties of Determinants

Section 5.1

Cofactors and Cramer’s Rule

Section 5.2

[4] Definitions and basic properties

[5] Examples

Week 15

Lecture 28 (April 29)

Eigenvalue and Eigenvectors

Section 6.1

[1] Example from my notes using properties of the determinant.

[2] Proof Cramer’s Rule (pg 250)

[3] An Eigenvalue Problem

[4] Definition on page 262

Applications of Eigenvalues and Eigenvectors

Section 6.3

[1] Compute the Powers of a matrix A

[2] Solve a second order linear relation with constant coefficients. Example 2 (pg 279)

Homework 14

Section 6.1

Problems:

[1] Problem 1a

Section 6.3

Problems:

[2] Problem 1

[3] Problem 7

Section 6.4

Problems:

[4] Problem 15

[5] Problem 21

[6] Problem 22

[7] Problem 24

Lecture 29 (May 2)

The Spectral Theorem and Quadratic Forms of a Matrix A

Section 6.4

[1] Example 1

[2] Theorem 4.1

[3] Example 4 (pg 290)

[4] Comments on two factorizations (see Example 3 on page 289)

Week 16

Lecture 30 (May 6)

Singular Value Decomposition (SVD): A generalization of Diagonalization for any rectangular matrix.

Applications of SVD to Pseudo-Inverse matrices, Dimensionality Reduction (Data Science PCA, Least Squares and Recommender Machines)

[1] My Example

Watch Professor Steve Brunton:

https://www.youtube.com/watch?v=yA66KsFqUAE

Student Learning Outcomes/Learning Objectives

Learning Objectives

The objectives of Linear Algebra are:

(1) The student will understand the mathematical concepts and terminology involved in Linear Algebra.

(2) The student will gain an acceptable level of computational proficiency involving the procedures in Linear Algebra.

(3) The student will understand the axiomatic structure of a modern mathematics subject and learn to construct simple proofs.

(4) The student will be able to apply his or her knowledge to applications of Linear Algebra.

The topics that will enable this course to meet its objectives are:

(i) the basic arithmetic operations on vectors and matrices, including inversion and determinants, using technology where appropriate;

(ii) solving systems of linear equations, using technology to facilitate row reduction;

(iii) the basic terminology of linear algebra in Euclidean spaces, including linear independence, spanning, basis, rank, nullity, subspace, and linear transformation;

(iv) the abstract notions of vector space and inner product space;

(v) finding eigenvalues and eigenvectors of a matrix or a linear transformation, and using them to diagonalize a matrix;

(vi) projections and orthogonality among Euclidean vectors, including the Gram-Schmidt orthonormalization process and orthogonal matrices;

(vii) the common applications of Linear Algebra, possibly including Markov chains, areas and volumes, Cramer's rule, the adjoint, and the method of least squares;

(viii) the nature of a modern mathematics course: how abstract definitions are motivated by concrete examples, how results follow from the axiomatic definitions and are specialized back to the concrete examples, and how applications are woven in throughout. This course will present various "characterization" theorems (eg. characterizing isomorphic finite-dimensional vector spaces by their dimension and characterizing invertible matrices by various criteria);

(ix) basic proof and disproof techniques, including mathematical induction, verifying that axioms are satisfied, standard "uniqueness" proofs, proof by contradiction, and disproof by counterexample;

(x) some of the common notions of higher mathematics, possibly including permutations, equivalence relations, the Kronecker delta function, canonical forms, and numerical techniques.