MATH 111 Linear Algebra

Fall 2010

Instructor: CHENG Shiu Yuen, Rm 3452. e-mail: macheng@ust.hk, Tel #: x7411 Office hour: Tu Th 1:45 pm2:45 pm at Rm 3452 or by appointment Course Description: Concepts, techniques and languages of Linear Algebra are essential for mathematical treatment of problems arising in science and engineering. A firm grasp of concepts and techniques in this course is then essential for science and engineering majors. This is also an excellent course for introducing mathematical proofs and algorithms. I intend to use this course to introduce students to proofs. Therefore, you are expected to do proofs. You will see a lot of them in lectures, homework assignments, and examinations.

Intended learning outcomes (ILO): Upon the completion of this course, students should be able to: 1. Develop an understanding of the core ideas and concepts (Gaussian elimination, vectors, matrices,

linear spaces, linear transformations, linear dependence and independence, basis, dimension, coordinates, matrix similarity, eigenvalues and eigenvectors, diagonalization, inner product, orthogonality, orthogonal projection) of Linear Algebra

2. Be able to recognize the power of abstraction and generalization in Linear Algebra (such as from Euclidean Space to Linear Space, from matrix to linear transformation, from dot product to inner product)

3. Be able to apply rigorous, analytic, highly numerate approach to analyze and solve problems using Linear Algebra

4. Be able to communicate problem solutions using correct mathematical terminology and good English.

Assessment of ILOs: There are various instruments listed in the following to assess the ILOs. Midterm Examination (No make-up exam): 6:00 pm-7:30 pm, Monday, 18 October, LTB Final Examination Project: Students are expected to form groups of four (special permission is needed for deviations)

to study and learn a section in the text not covered in class. Each group has to submit a written report on Thursday, 25 November, and make an oral presentation on Sunday, 28 November. Details of the project will be announced in October.

Text: Linear Algebra and Its Applications, 3rd edition, by David Lay, Addison Wesley Syllabus: Except for a few sections, I intend to cover from Chapter 1 to Chapter 6 in the text A few topics will be added if time permits. Students can download all the transparencies of the lectures at the website http://www.laylinalgebra.com (click on the original version). Grading Policy: The course grade will be assigned according to an absolute scale in the following.

F: 0-54; D range: 5459; C range: 6069; B range: 7084; A range: 85100 The course grade is computed from three assessment instruments: Project (P), Midterm Examination (M), Final Examination (FE). P counts 20%, M counts 30%, and FE counts 50% towards the course grade. Homework assignments will be assigned regularly but will not be collected. On average, there are about 8 assigned problems per lecture. The assignments are intended to help you learn the materials covered in the lecture. You are strongly advised to complete the assignment every week. You cannot learn the materials if you do not work on the assignments. At least 60% of the midterm and final exam questions are based on the homework assignments and examples in the lectures. The list of all assignments from the text is attached.

Intended learning outcomes: Upon the completion of this course, students should be able to:

1.Develop an understanding of the core ideas and concepts (Gaussian elimination, vectors, matrices, linear spaces, linear transformations, linear dependence and independence, basis, dimension, coordinates, matrix similarity, eigenvalues and eigenvectors, diagonalization, inner product, orthogonality, orthogonal projection) of Linear Algebra.

2.Be able to recognize the power of abstraction and generalization in Linear Algebra (such as from Euclidean Space to Linear Space, from matrix to linear transformation, from dot product to inner product)

3.Be able to apply rigorous, analytic, highly numerate approach to analyze and solve problems using Linear Algebra.

4.Be able to communicate problem solutions using correct mathematical terminology and good English.

Grading Policy: The course grade will be assigned according to an absolute scale in the following.

F: 0-54; D range: 5459; C range: 6069; B range: 7084; A range: 85100

Homework assignments will be assigned regularly but will not be collected. The course grade is computed from three components: Project (P), Midterm Examination (M), Final Examination (FE).

P counts 20%, M counts 30%, and FE counts 50% towards the course grade.

The midterm examination is scheduled in the evening of Monday, 18 October, from 6:00 pm to 7:30 pm. There is no make-up examination for the midterm examination

Students are expected to form groups of four (special permission is needed for deviations) to study and learn a section in the text not covered in class. Each group has to submit a written report on 25 November and make an oral presentation on Sunday, 28 November. Details of the project will be announced in October.

Academic integrity is most important and you are expected to observe the university guidelines and regulations (Academic Calendar p. 26-p. 28). Any violations will be dealt with strictly according to the guidelines and regulations.

MATH 111 Linear Algebra

Fall 2010 Assignment from the text

Section1.1: 7, 9, 11, 17, 21, 25, 29, 31 Section 1.2: 3, 6, 11, 15, 19, 29, 30, 31 Section 1.3: 9, 17, 21, 22, 26, 29, 32 Section 1.4: 9, 13, 16, 19, 26, 31, 32, 33, 34, 37, 39, 41 Section 1.5: 11, 13, 17, 25, 26, 39, 40 Section 1.7: 5, 7, 8, 11, 39 Section 1.8: 3, 11, 19, 24, 25, 31, 34, 36 Section 1.9: 7, 15, 25, 35, 36

Section 2.1: 11, 19, 21, 22, 23, 24, 39 Section 2.2: 12, 13, 15, 16, 21, 22, 29, 31, 33, 34, 35 Section 2.3: 5, 7, 13, 15, 17, 24 28, 33, 36, 37, 38 Section 2.5: 3, 5 Section: 2.8: 15, 17, 19, 23, 25, 31, 33, 37 Section 2.9: 3, 5, 9, 11, 13, 14, 21, 26, 29

Section 3.1: 47 9, 13, 38 Section 3.2: 5, 7, 19, 25, 29, 31, 32, 34, 35, 36

Section 4.1: 5, 8, 11, 15, 16, 21, 26, 27, 28, 29, 32, 33, 34 Section 4.2: 5, 15, 28, 30, 33, 35, 36, 38 Section 4.3: 9, 13, 14, 15, 19, 23, 29, 30, 31, 32 Section 4.4: 5, 7, 9, 10, 25, 33, 35, 36 Section 4.5: 11, 13, 17, 21, 23, 26, 31, 32 Section 4.6: 1, 2, 3, 5, 9, 13, 15, 21, 23, 25, 26, 33 Section 4.7: 7, 13, 19

Section 5.1: 13, 15, 17, 19, 23, 25, 26, 27 Section 5.2: 3, 11, 17, 18, 19, 20, 23, 24 Section 5.3: 5, 9, 11, 13, 19, 27, 28 Section 5.4: 5, 15, 19, 20, 21, 22, 24, 25, 26

Section 6.1: 1, 3, 5, 9, 24, 27, 29, 30, 31 Section 6.2: 7, 9, 11, 17, 21, 26, 27, 28, 29, 33 Section 6.3: 7, 9, 11, 13, 15, 19, 24 Section 6.4: 9, 11, 13, 15 Section 6.5: 9, 11, 19, 20, 22, 23 Section 6.7: 3, 11, 13, 14, 15, 16, 17, 18, 25

Example: Let 1( , , )nf x x be a 2C function. Then, the Taylor expansion of

f at the point 1( , , )na a is of the form

23

1 11 , 1

( , , ) ( , , ) ( ) ( ) ( ) ( )( ) ( )n n

n n i i i i j ji i ji i j

f ff x x f a a a x a a x a x a xx x x

= =

= + + +

The method of Gaussian elimination appears in Chapter Eight, Rectangular Arrays, of the important Chinese mathematical text Jiuzhang suanshu or The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations. The first reference to the book by this title is dated to 179 CE, but parts of it were written as early as approximately 150 BCE. It was commented on by Liu Hui in the 3rd century.

However, the method was invented in Europe independently by Carl Friedrich Gauss when developing the method of least squares in his 1809 publication Theory of Motion of Heavenly Bodies.