Margaret
Menzin Office:S209
Phone: X2704
Email:
menzin@simmons.edu
Home Phone: 781-862-5107
Office Hours: MWF 7:15-8:00; and
M 10:00-4:30;
W 10:00-3:30
Fri 10:00-11;30;
12:30-1:30 and often later
Note: I am also available some Wednesdays after 3:30 and
sometimes
later on Fridays.
The
Mathematics, Statistics and Computer Science Department eats at Bartol on Fridays
at
11:30. We hope you will join us.
Text: Rorres and Anton: Elementary Linear Algebra Applications Version 9th Edition
Software: We will use Sage, which is in all the labs and is available for free download at http://www.sagemath.org/
It is critical that you read the directions before you download the software or use it on-line.
This course has three meetings in a classroom. Absence from class is occasionally unavoidable, but is strongly discouraged.
At the start of each class I usually ask if there are questions (including on homework). If I forget to do so, please feel free to ask question. I bring a copy of our text to class, so you do not need to do so. You should, however, make a note for yourself about which problems to ask about. If you have a question about material in the chapter it is most helpful if you can be specific about at what page and paragraph the book lost you.
It is critical that you do homework as assigned. Problems always look do-able, but when you start to do them, and compare your work with the answers in the back you may uncover gaps in your understanding. You should spend about 3 hours on homework for every hour of class time.
Students with Disabilities: Reasonable accommodations will be provided for students with documented physical, sensory, systemic, cognitive, learning, and psychiatric disabilities. If you have a disability and anticipate that you will need a reasonable accommodation in this class, it is important that you contact the Academic Support Center Director at 617.521.2471 early in the semester. Students with disabilities receiving accommodations are also encouraged to contact their instructors within the first 2 sessions of the semester to discuss their individual needs for accommodations."
Software As noted above, we will be using Sage for our calculations in this course. Sage is a brand new open source, free package for mathematical calculations. I am very excited about this as, for the first time, you will be able to put free powerful mathematical software on your own computer. I have also asked to have Sage be put on the computers in S251A and S241, so it should be readily available. The availability of Sage means that we can focus on theory and applications and lighten the calculational burden: exciting news indeed!
You are about to start the study of linear algebra: a subject which includes aspects of algebra (systems of linear equations), geometry (lines and planes, as well as rotations, reflections and projections), mathematical objects of great utility in science (vectors and matrices), abstract mathematical objects (vector spaces) with structure (vector addition and scalar multiplication), functions which preserve that structure (linear transformation - realized for finite dimensional vector spaces, such as vectors in the plane, by matrix multiplication) and last, but certainly not least, applications to both mathematics (especially to solving differential equations and to linear programming) and other fields (from economics to physics). In short, this is a wonderful subject - both elegant and useful.
The course is broken into four units. There is a take home test at the end of each unit. I will give you some warning as to when the test will be handed out. You will have a week to complete the test. All tests are open book.
While exams in this course are not explicitly
cumulative, the material in the course builds on itself.
If you do not understand
a topic it behooves you to ask about it so that you can use the material in later parts of the course.
I mentioned earlier that linear algebra is an area rich with applications. Most applications are found in Chapter 11 of your text. Some applications will be woven into the first three units of the course. Other applications require all the material in the first three units and will be examined in Unit 4.
Your text also has a number of technology exercises. The technology exercises for all sections in a chapter are found at the end of the chapter and are designated by section as T1, T2, etc. So, when the list of problems for a section includes a problem numbered Tx, you need to turn to the end of the chapter to find the problem.
Goals for the Course
1. Students will be able to solve small systems of linear equations by hand and larger ones with computer-based tools, and be able to
explain why they got a certain dimension solution set and to interpret the results.
2. Students will understand the concepts of vector spaces, subspaces, dimension, basis, linear independence, inner product, and be able to prove theorems about these concepts and to give
concrete examples illustrating various situations.
3. Students will be understand the concepts of linear transformations, matrices, rank, kernel, orthoganality, eigenvalues
and be able to explain in writing how they are related. They will be able to prove basic theorems which use these concepts and to use examples from low dimensional vector spaces to
illustrate these ideas.
4. Students will be able to apply linear algebra to other mathematical problems, such as modeling a simple Markov processes or solving a set of homogeneous differential equations.
5. Students will be able to use Sage for matrix computations and be able to explain in writing the
limitations of Sage or any similar software package.
Unit 1: Systems of Linear Equations and Matrices Chapters 1 and 2 of your text
We begin by investigating what happens when we try to solve a system of linear equations
2x + 3y = -1
3x - 2y = 5
Why do some systems have no solutions, some exactly one and some an infinite number? Is there a way we can routinize
finding the solution to such systems? What is the relationship between an arbitrary system of equations and
a system where all the constants on the right side of the equation are zero? Is there a neat way to describe the set
of solutions to a given system of equations? What is the geometrical interpretation?
This leads us to the study of matrices, determinants, and inverses of matrices.
Unit 2: Vector Spaces and Linear Transformations Chapters 3, 4 and 5 of your text
In Unit 1 we will have found that vectors may be used to describe the set of solutions to a system of
linear equations. Further, in the case where all the constants on the right are zero, we will have discovered
that the vectors which are solutions have nice properties --- the vector sum of two solutions is another solution,
a scalar multiple of two solutions is another solution. (We will also have a simple way to deal with the case
where the constants are not all zero.) This generalizes to the concept of a vector space.
After a brief review of vectors in the plane and in 3-dimensional space we move to consideration of
vectors in n-dimensional space. The 'nice' functions between such vector spaces are called linear transformations.
(These are the functions which preserve, as you will see, our structure of vector addition and scalar multiplication.)
These functions may all be written as multiplication by a matrix, and there are elegant connections between functions
and their matrices. For example, if a linear transformation T is multiplication by the matrix A, and A has an inverse,
then the inverse of A is the matrix for the linear transformation T -1. We will explore these connections.
Our knowledge of simple geometric transformations, such as rotating about the origin, will provide us with examples and
insight as we explore linear transformations and matrices.
Finally, we turn our attention to general vector spaces. This is the first set of formal mathematical objects
you have seen, and many of the patterns we will see (and note) here will reappear in all the abstract mathematics
you see in the future. Vector spaces are not difficult, and seeing how the formal pieces fit together will
make other math courses much easier.
There are many vector spaces beyond n-dimensional space.
For example, the set of all polynomials in X forms a vector space.
So, we ask how we may describe all vector spaces, and which ones are essentially the same. (Think of congruent triangles, which
are essentially the same.)
This brings us to the enormously important concepts of dimension,
via the concepts of a basis for a vector space, linear independence and spanning.
Unit 3: More on Bases of Vector Spaces and Linear Transformations, including Eigenvectors
Chapters 6, 7 and 8 of your text
Leaning on the concepts or inner product (which generalizes the dot product of vectors in
2 and 3 dimenesional space) and orthogonality (which generalizes the notion of perpendicularity) and
length, we ask if there are any particularly nice bases for a vector space (as unit vectors along the X,Y
and Z axes are for 3-dimensional space.) The Gram-Schmidt Orthogonalization Process (which is also very
useful in chemistry) answers our needs.
Next we ask if, when we look at a linear transformation (i.e. a function from one vector space to another) if
there is a basis for the domain which is particularly useful. For example, if we reflect about the line x = y in
the plane, then anything along that line stays the same and anything along the line x = -y is mapped to a negative
of itself. This leads us to the incredibly important notion of eigenvectors and eigenvalues. Eigenvectors
are important for both theory and applications and we will spend some time in the next unit
looking at the applications.
Unit 4: Advanced Applications Chapters 9 and 11 of your text
We will first look at the application of linear algebra to solving linear differential
equations. For example, we will discover why the solutions to simple harmonic motion all
look like y = A cos rx + B cos rx. We will also examine Markov processes and some applications
to a variety of economic, biological, and mathematical problems. Finally, we will look at some
linear programming problems and applications to management problems.
Section 2.1 #1, 2a, 3bc, 6a, 9 and verify with Sage, 12 and verify with Sage,
16, 17 and verify solution with Sage, 25, 26, 29, (optional 33), T1, t2
Section 2.2 #1a, 2b, 3ab, 4, 14, 15b, 18, 20
Section 2.3 #1a, 4cd, 6, 8, 9, 12a, 14b, 15b, 16, 17, 18, 20, 23
Section 2.4 -Browse only
Supplement for Ch. 2 #(optional 5), 9, (optional 10), 15
Applications: While you are working on your take-home test I will discuss sections 11.7, 11.9 and 11.10
Chapter 3 is a review of vectors in R2 and R3
so we will move through it quickly.
Section 3.1 #1ab, 2ahi, 6b, 8, 11a, 15, 21
Section 3.2 #1ae, 2ab, 3a, 4, 9, 10, 15, 17
Section 3.3 #1c, 2c, 3, 4ad, 6c, 8, 9, 10, 13, 16ab, 21, 27, 30
Section 3.4 #1ab, 20, 25, 35a, 38
Section 3.5 Browse #1a, 7a, (optional 17), 47
Section 4.1 #1ab, 3, 6abef, 7, 9cd, 10, 11d, 17d, 21-24, 27, 34, 37
Section 4.2#1c, 3, 4, 8, 11, 15a, 18, 20, 26, 29, 31
Section 4.3 #1, 2ad, 3-5, 7, 9, 11, 12, 14-16, 18, 20, 23-24, 26, 27
Section 4.4 Browse only
Section 5.1 #2, 3, 7, 10-14, 17c, 18-20, 27, 29, 31
Section 5.2 #1-5, 6c, 8a, 11, 17, 21, 23, 24, 27, T1
Section 5.3 # 1ad, 2bd, 3c, 4c, 6, 10, 12, 13, 15, 21, 25, T1
Section 5.4 #1, 2, 6, 7ab, 10, 13, 18, 20, 21b, 23, 24, 29a, 32
Section 5.5 #1, 2ab, 3ab, 4, 5ad, 6ad, 7ab, 11b, 15, 16, 19, T1-T3
Section 5.6 #2ac, 4, 5, 7, 10, 12a, 16a, 17-19, T1
Supplement to Ch. 5 #1, 3a, 5, 6, 8, 10, 11, 13a
Section 6.1 #3, 4a, 8a, 10, 13, 16, 17, 0, 23, 26, 27a, 28a, 32, T2, (optional T3)
Section 6.2 #5ae, 6a, 11, 12ac, 15-17, 18ab, (optional 19), 21, 23, (optional 24), read 27-29, 30, 33, 35, 37
Section 6.3 (We won't worry about QR decomposition, but you should read the material.) #1, 2, 7ab, 9ab, 11, 17, (optional 29), 34, 37, 38, T1, T2
Section 6.4 Browse. What does this process do when you apply the least squares process to approximating u
first in W1, then in W2 etc. where
W1 ⊂ W2 ʹ W3 ⊂... ⊂ Wn
Section 6.5#1ab, 2ab, 3a, 6, 11, 12, 14, 15, 18
Section 6.6#4, 5, 6a, 13, 14, (optional 15 and 20)
Supplement to Ch. 6 #3, 12, 13
Section 7.1 #1abef, 2ab, 4ab, 10, 11, 17, 18, 20, 22, 24, T1, T2, T4, T5
Section 7.2 #1, 2, 7, 8, 15, 17, (optional do other problems between 12 and 16 on Sage), 22, 25, 26, T2
Section 7.3 #1abc, 2, (optional 8), 14
Supplement to Ch. 7 #1, 2, read 4, read 5, 7a, 13, 14
Section 8.1 #2-7, 10, 12, 15, 17ab, 19, 20, 22, 27, 29-34
Section 8.2 #1, (optional 4), 5, 7ac, 8ac, 9ac, 10 (note instructions are on previous page), 14, 15, 16, 17, 25-30
Section 8.3 #1abcd, 3ac, 5, 6, 7, 9, 10ac, 11, 12, 17ab, 19 and explain why this is important, 20-25, 28
Section 8.4 #1, 4, 5, 9, 11, 13, 15, 19-23; Note: this section is calculations you need to be able to do for the next section
Section 8.5 #1, 3, 9, 11, 13, 15, 19-23
Section 8.6 #2, 5-7, 10 (use what you learned in Discrete Math!), (optional 11), 12
Supplement to Ch. 8 #2-5, 9, 10, 14, 19, 22, (optional 23 and 25)
Section 9.1 #1, 3, 6, 7, 8, 10
As time allows:
Section 9.3 #1, 2, 4
Section 9. 4 #1, 4, 10
Section 9.8: Browse
Section 11.6 #1, 2, 3c, (optional 7)
Section 11.18
Section 11.3
As time allows:
Section 11.8 #5a
Section 11.19
Section 11.17