## Instructor: Dr. D. A. Kouba

Last Updated: February 10, 2019

(This website is currently under construction.)

Text: Elementary Linear Algebra (11th edition) Applications Version by Anton and Rorres

Office: 3135 MSB

Phone: (530) 601-4444x2007

Regular Office Hours: Variable (announced in class and sent via e-mail) or by appointment

• Here is a list of Exam ID Numbers .

 TA OFFICE HOURS Wednesday 9:45-10:45 a.m. Cooper Jacob 2137 MSB 2nd Office Hour Variable Cooper Jacob 2137 MSB

EXAM DATES :

• EXAM 1-- FRIDAY, April 20, 2018

• EXAM 2-- FRIDAY, May 11, 2018

• EXAM 3-- FRIDAY, June 1, 2018

• FINAL EXAM -- MONDAY, June 11, 2018, 3:30-5:30 p.m. in 106 Wellman

The course will likely cover the following sections in our textbook : 1.1-1.7, 2.1, 2.2, 3.1-3.3, 4.1-4.5, 4.7, 4.8, 5.1, 5.2, 9.1

SOLUTIONS TO ALL HOMEWORK ASSIGNMENTS AND HOUR EXAMS CAN BE VIEWED ON THE INTERNET at Math 22A Homework and Exam Solutions . In addition, you may look at a copy of solutions during my office hours (or appointment) in 3135 MSB.

Here is a copy of the Course Syllabus .

Here is a copy of the Schedule of Lectures .

Here are copies of Supplementary Class Handouts .

The following TIPS were originally written for Calculus students, but they apply equally well to Linear Algebra students.

### These are the OPTIONAL EXTRA CREDIT Survey and Short Paper .

The following homework assignments are subject to minor changes.

SCANNED PROBLEMS for Sections 1.1-1.7

• HW #1 ... (Section 1.1) ... Here are some worked out Examples solving two-by-two Linear Systems of Equations ... Here are the rules for Matrix Row Reduction using Elementary Row Operations ... p. 8: 1, 5, 7, 8, 9abcd, 11, 12, 14abc, 16-20, 24, 25, 26 (Solve the system using elementary row operations.), 27, and True-False

• HW #2 ... (Section 1.2) ... p. 22: 3abd, 4abd, (Use Gaussian elimination or Gauss-Jordan elimination for the following 3 problems.) 5-7, 16, 17, 19, 20, 23, 26 (CHANGE the equation "x+2y-(a^2-3)z=a" TO "x+2y+(a^2-a+1)z=a+1".), 27, 29, 30, 33, 36 (Change the numbers 1, 0, and 5 on the right-hand side to -5, 8, -11 to get nicer answers.), 37, 39, 41, 43, and True-False

• HW #3 ... (Section 1.3) ... p. 36: 1, 2, 4, 7, 9a, 11, 13b, 16, 18, 19, 23, 24, 25, 26bd, 27-30, 32b, 33, 36a, and True-False

• HW #4 ... (Section 1.4) ... Here are some basic facts about Zero Matrices, Matrix Arithmetic, and Transpose Arithmetic ... p. 49: 2c, 3b, 5, 7, 9-12, 15, 17, 18, 20ab, 23, 24, 31, 32, 33a, 34-37, 40, 41, 43a, 45, 46, 50, and True-False

• HW #5 ... (Section 1.5) ... p. 58: 1-3, 5, 8-10, 11a, 12b, 13, 15, 18, 19b, 20b, 21, 22, 29, 30, 32, and True-False

• HW #6 ... (Section 1.6) ... Here is a link with a brief explanation (courtesy of Scott Gregory at the University of Washington) about how the inverse of a matrix can be used in Encryption ... Here is a link with a brief explanation (courtesy of Ting Yip at the University of Washington) about how matrix multiplication (for Translations and Rotations) is used in Computer Graphics ... Here I verify why Rotation Matrices and Translation Matrices work the way they do ... p. 66: 1, 4-7, 9, 12-15, 17, 18b, 20, 22, 23 (Ignore the statement: Prove that every matrix of this form is a solution.), and True-False

• HW #7 ... (Section 1.7) ... Here are brief notes and some random practice problems regarding Diagonal, Triangular, and Symmetric Matrices ... p. 72: 7, 10, 12-14, 19-21, 25-27, 30-32, 34, 36, 39, 47, and True-False
EXAM 1 is Friday, April 20, 2018. It will cover handouts, lecture notes, and examples from class, homework assignments 1 through 7, and material from sections 1.1-1.7 in the book which was presented in lecture notes through Wednesday, April 18, 2018. MOST of the exam questions will be homework-type and lecture example-type problems.

TYPES OF QUESTIONS FOR EXAM 1 FOR SPRING 2018 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)

• 3-- Solve Linear Sytems of Equations Using Any Method of Using Gauss-Jordan Method (Reduced Row Echelon Form)
• 2-- Find Inverse of a Matrix
• 2 or 3-- True or False (Prove or Disprove)
• 1 or 2 -- Others
• 1 -- OPTIONAL EXTRA CREDIT

HERE ARE SOME RULES FOR EXAM 1.

• 0.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM ANOTHER STUDENT'S EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO HAVE ANOTHER PERSON TAKE AN EXAM FOR YOU. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
• 1.) No notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM.
• 2.) You will be graded on proper use of limit notation.
• 3.) You will be graded on proper use of derivative and integral notation.
• 4.) Put units on answers where units are appropriate.
• 5.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

SOLUTIONS TO EXAM 1 ARE POSTED ON THIS WEBPAGE.

THE GRADING SCALE FOR SPRING 2018 EXAM 1 IS :

A+ ...... 100-108

A ...... 90-99

A-/B+ ...... 86-89

B ...... 70-85

C ...... 50-69

D ...... 40-49

F ...... 0-39

SCANNED PROBLEMS for Sections 9.1, 3.1-3.3, 2.1, 2.2, 4.1, 4.2

• HW #8 ... (Section 9.1) ... Here are worked out examples and other notes on Solving Systems Using LU-Decomposition ... The introduction of LU-Decomposition into academia and industry in the 1940's is credited to Alan Turing , considered by many to be the "Father of Computer Science" ... p. 499: 1-7, 9ab (optional), 10a, 13 (optional), 18, and True-False (a), (b), (c) only

• HW #9 ... (Section 3.1) ... Here is a list of properties for Vector Algebra and information about the Geometry of Vectors ... p. 140: 1-3, 6, 8, 9, 10a, 11b, 14, 15, 16, 18-20, 22-27, and True-False

• HW #10 ... (Section 3.2) ... Here is a list of properties for the Dot Product and Norms of vectors ... p. 153: 1a, 2b, 3, 7, 9, 11, 13-16, 18, 19b, 21-23, 24a, 26, 27, 30, 32a, and True-False

• HW #11 ... (Section 3.3) ... p. 162: 1, 2bc, 3-5, 7-9, 11, 12, (For problems 13 and 14 find each projection vector AND it's magnitude.), 15-17, 20, 22, 23, 26, 27, 29-31, 33, 34, 36-38, and True-False

• HW #12 ... (Section 2.1) ... Here are some notes on Minors, Cofactors, and Determinants ... It's not clear why computing the determinant of matrix A using cofactor expansion along any row or column gives the same answer. Scroll down to Theorem (Cofactor expansion) to see a detailed Proof of this fact ... p. 111: 1, 3ac, 4c, 5, 6, 8-10, 12, 15-17, 19cd, 21, 23, 28, 29, 33-35, 38, and True-False

• HW #13 ... (Section 2.2) ... Here are some notes on computing Determinants Using Row Reduction ... p. 117: 2, 3, 6, 7, 9, 14-18, 21, 22, 24, 29, 30, 35, and True-False ... and ... (Section 2.3) ... p. 117: 2, 4, 5, 8, 11, 15-18, 30 (Show only that A is invertible.), 34-37, and True-False (abcdij only)

• HW #14 ... (Section 4.1) ... Here are some notes on Real Vector Spaces ... p. 190: 1, 2, 4, 5, 7, 8, 12, 17, 28, and True-False

• HW #15 ... (Section 4.2) ... p. 200: 1, 2abe, 3b, 4b, 5c, 7ab, 8b, 9ac, 10a, 11, 12ac, 13, 14, 15abc, 17, 19b, 20a, and True-False
EXAM 2 is Friday, May 11, 2018. It will cover handouts, lecture notes, and examples from class, homework assignments 8 through 15, and material from sections 2.1, 2.2, 3.1-3.3, 4.1, 4.2, and 9.1 in the book which was presented in lecture notes through Wednesday, May 9, 2018. MOST of the exam questions will be homework-type and lecture example-type problems.

TYPES OF QUESTIONS FOR EXAM 2 FOR SPRING 2018 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)

• 1 -- TRUE or FALSE
• 1 -- Proof of Theorem
• 1 -- Compute Determinant
• 2 -- Problems Regarding Lines and Planes in 3-Space
• 1 -- LU-Decomposition Problem
• 1 -- Vector Space/Subspace Problem
• 1 -- Other
• 1 -- OPTIONAL EXTRA CREDIT

HERE ARE SOME RULES FOR EXAM 2.

• 0.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM ANOTHER STUDENT'S EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO HAVE ANOTHER PERSON TAKE AN EXAM FOR YOU. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
• 1.) No notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM.
• 2.) You will be graded on proper use of derivative and integral notation.
• 3.) Put units on answers where units are appropriate.
• 4.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

SOLUTIONS TO EXAM ARE POSTED ON THIS WEBPAGE.

THE GRADING SCALE FOR SPRING 2018 EXAM 2 IS :

A+ ...... 100-110

A ...... 90-99

A-/B+ ...... 86-89

B ...... 68-85

C ...... 48-67

D ...... 35-47

F ...... 0-34

SCANNED PROBLEMS for Sections 4.3-4.8, 6.3, 6.4

• HW #16 ... (Section 4.3) ... Here are some notes on the Wronskian ... p. 210: 1, 2, 3a, 4a, 5-7, 8ac, 9a, 10b, 11-13, 16abce (Use trig identities or a Wronskian.), 18-22, 24 (Do sets {v2, v3} and {v1} only.), 26 (Change "P2" to "P1" and change "more than three vectors" to "exactly three vectors".), 28, 29, and True-False

• HW #17 ... (Section 4.4) ... Here are some notes on Spanning Sets ... p. 219: 1 (Use a determinant.), 2 (Use a determinant.), 3 (Show that a+bx+cx^2 can be written as a linear combination of the given polynomials. Show that the given polynomials are linearly independent.), 7a (Use a determinant.), 8 (Show that the given polynomials do not SPAN P2 OR show that the given polynomials are linearly DEPENDENT.), 9, 10, 14, 20, 29, 31, and True-False

• HW #18 ... (Section 4.5) ... Here are some (hard to read) handwritten notes on the Dimension of Vector Spaces ... p. 228: 1-3, 6, 7abc, 8ab, 9ab (Change nxn to 3x3 and remove the word "diagonal".), 10, 12-14, 16-18, 20a, 21a (Use n=3.), 25, 27, and True-False

• HW #19 ... (Section 4.7) ... p. 246: 1a, 2b, 3, 4b, 5, 7, 8b, 9a, 10a, 11, 12b, 13a, 14-16, 18, 20, 21b, 23, 24 (For part b find an example different from a coordinate axis. For part c find an example different from the coordiante planes.), 25 (For part b do A, C, and D only.), 27, 28, and True-False

• HW #20 ... (Section 4.8) ... p. 256: 1, 2a, 3-8, 10-21, 24, 28-30, 33, and True-False

• HW #21 ... (Section 6.3) ... Here are some notes on the Gram-Schmidt Process ... p. 376: 1abc, 2ac, 3, 4b, 7, 9, 15, 18, 21-23, 26-29, 32, 33, 43, 55 (Optional) and True-False (parts (c) and (e) only)

• HW #22 ... (Section 6.4) ... p. 386: ... Here are some notes on how to find Least Squares Solutions to unsolvabe systems of equations ... (Find the Least Squares Solution and the Least Squares Error for problems 3-5, 12, and 13) ... 3-5, 12, 13, 16, 17, 19, 21, 22, 26, and True-False
EXAM 3 is Friday, June 1, 2018. It will cover handouts, lecture notes, and examples from class, homework assignments 16 through 22, and material from sections 4.3-4.5, 4.7, 4.8, 6.3, and 6.4 in the book which was presented in lecture notes through Wednesday, May 30, 2018. MOST of the exam questions will be homework-type and lecture example-type problems.

Bring it ...

TYPES OF QUESTIONS FOR EXAM 3 FOR SPRING 2018 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)

• 2 -- Determine if Vectors are Linearly Independent or Linearly Dependent
• 1 or 2 -- Prove Set is a Subspace
• 1 or 2 -- Orthogonal Complement Problems
• 1 -- Row Space, Column Space, Null Space Problem
• 1 -- Gram-Schmidt Problem with Three Vectors
• 1 -- Rank, Nullity Problem/Proof
• 1 -- Least Squares Problem
• 1 or 2 -- Others
• 1 -- OPTIONAL EXTRA CREDIT

HERE ARE SOME RULES FOR EXAM 3.

• 0.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM ANOTHER STUDENT'S EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO HAVE ANOTHER PERSON TAKE AN EXAM FOR YOU. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
• 1.) No notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM.
• 2.) You will be graded on proper use of derivative and integral notation.
• 3.) Put units on answers where units are appropriate.
• 4.) Do not use any shortcuts from the book when using the method of integration by parts.
• 5.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

SOLUTIONS TO EXAM 3 ARE POSTED ON THIS WEBPAGE.

THE GRADING SCALE FOR SPRING 2018 EXAM 3 IS :

A+ ...... 100-110

A ...... 90-99

A-/B+ ...... 84-89

B ...... 65-83

C ...... 45-64

D ...... 34-44

F ...... 0-33

SCANNED PROBLEMS for Sections 5.1, 5.2

• HW #23 ... (Section 5.1) ... Eigenvectors are used in many applications. They are used when you do a Google Search , as is nicely explained here by 278Brandon. In the context of wildlife biology, Stable-Age Distributions are determined by the eigenvectors of Leslie Matrices , which use the reproductive rates and and survival rates of species with discrete (seasonal) breeding periods ... p. 300: 1, 2, 4, 5, 6a, 7, 10, 12, 24, 25, 27, 33, 34, 38, 39, and True-False

• HW #24 ... (Sections 5.2) ... p. 311: 1, 4, 5, 8, 17, 20ad

The FINAL EXAM is Monday, June 11, 2018

3:30-5:30 p.m.

in 106 Wellman

BRING A PICTURE ID TO THE EXAM
AND BE PREPARED TO SHOW IT TO KOUBA OR THE TEACHING ASSISTANT !!

The final exam will cover handouts, lecture notes, and examples from class, homework assignments 1 through 24, material from sections 1.1-1.7, 2.1, 2.2, 3.1-3.3, 4.1-4.5, 4.7, 4.8, 5.1, 5.2, 9.1.

TYPES OF QUESTIONS FOR THE FINAL EXAM FOR SPRING 2018 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.).

• 1 -- LU-Decomposition Problem
• 1 -- Gram-Schmidt Problem
• 1 -- Least Squares Solution and Least Squares Error Problem
• 1 -- Diagonalization Problem
• 1 or 2 -- Eigenvalue/Eigenvector Problem
• 3 or 4 -- True or False (Prove or Disprove) Problems
• 1 -- Subspace Problem
• 1 or 2 -- Others
• 1 -- OPTIONAL EXTRA CREDIT

HERE ARE SOME RULES FOR THE FINAL EXAM.

• 0.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM ANOTHER STUDENT'S EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO HAVE ANOTHER PERSON TAKE AN EXAM FOR YOU. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
• 1.) No notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM.
• 2.) You will be graded on proper use of limit notation.
• 3.) You will be graded on proper use of derivative and integral notation.
• 4.) Put units on answers where units are appropriate.
• 5.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

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