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Math 130 Linear Algebra
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Please bookmark this page, http://aleph0.clarku.edu/~ma130/, so you can readily access it.
General information
- Course description.
Math 130 is a requirement for mathematics and physics majors, and it’s highly
recommended for majors in other sciences especially including computer-science
majors. Topics include systems of linear
equations and their solutions, matrices and matrix algebra, inverse matrices;
determinants and permutations; real n-dimensional vector spaces, abstract vector
spaces and their axioms, linear transformations; inner products (dot products),
orthogonality, cross products, and their geometric applications; subspaces,
linear independence, bases for vector spaces, dimension, matrix rank;
eigenvectors, eigenvalues, matrix diagonalisation. Some applications of linear
algebra will be discussed, such as computer graphics, Kirchoff’s laws,
linear regression (least squares), Fourier series, or differential equations.
See also
Clark’s Academic Catalog.
- Prerequisites.
The prerequisite for the course is one year of college calculus, others by permission only.
- Office hours and tutoring hours
Prof. R. Broker. Thursday 4:00–5:00 and by appointment. Room BP 345
Prof. E. Joyce. MWF 10:00–10:50, MWF 1:00–2:00. Room BP 322
K. Schultz. Tutoring Monday 8:00–10:00. Room BP 316
- Assignments & tests.
There will be numerous short assignments, mostly from the text, occasional quizzes,
two tests during the semester, and a two-hour final exam during finals week in December.
The two tests during the semester are yet to be scheduled.
- Time and study.
Besides the time for classes, you'll spend time on reading the text, doing the assignments, and studying of for quizzes and tests. That comes to about five to nine hours outside of class on average per week, the actual amount varying from week to week. Here's a summary of a typical semester's 180 hours
Regular class meetings, 14 weeks, 42 hours
Two evening midterms and final exam, 6 hours
Reading the text and preparing for class, 4 hours per week, 56 hours
Doing weekly homework assignments, 56 hours
Meeting with tutors or in study groups, variable 4 to 12 hours
Reviewing for midterms and finals, 12 hours
- Course grade. The course grade will be determined as follows:
2/9 assignments and quizzes,
2/9 each of the two midterms, and
1/3 for the final exam.
- Matlab. There are several different symbolic mathematics programs. We’ll use the one called Matlab. A couple of others you may have heard of are Maple and Mathematica. They can be used to perform various mathematical computations. You’ll need to know how to do these computations and perform small computations by hand, but for large computations, it helps to have a program do them to save time and reduce mistakes.
- Course goals.
- To provide students with a good understanding of the concepts and methods of linear
algebra, described in detail in the syllabus.
- To help the students develop the ability to solve problems using linear algebra.
- To connect linear algebra to other fields both within and without mathematics.
- To develop abstract and critical reasoning by studying logical proofs and the
axiomatic method as applied to linear algebra.
- LEEP learning outcomes.
- Knowledge of the Natural World and Human Cultures and Societies—including foundational disciplinary knowledge and the ability to employ different ways of knowing the world in its many dimensions. Students will
- develop an understanding of linear algebra, a fundamental knowledge area of mathematics
- develop an understanding of applications of linear algebra in mathematics, natural, and social science
- develop an appreciation of the interaction of linear algebra with other fields
- Intellectual and Practical skills—including inquiry and analysis, the generation and evaluation of evidence and argument, critical and creative thinking, written and oral communication, quantitative literacy, and information literacy. Students will
- be able to employ the concepts and methods described in the syllabus
- acquire communication and organizational skills, including effective written communication in their weekly assignments
- be able to follow complex logical arguments and develop modest logical arguments
- Personal and Social Responsibility—including ethical reasoning and action, the intercultural understanding and competence to participate in a global society, civic knowledge and engagement locally as well as globally, and the lifelong habits of critical self-reflection and learning. Students will
- begin a commitment to life-long learning, recognizing that the fields of mathematics, mathematical modeling and applications advance at a rapid pace
- learn to manage their own learning and development, including managing time, priorities, and progress
- Ability to Integrate Knowledge and Skills—including synthesis and advanced accomplishment across general and specialized studies, bridging disciplinary and interdisciplinary thinking, and connecting the classroom and the world. Students will
- recognize recurring themes and general principles that have broad applications in mathematics beyond the domains in which they are introduced
- understand the fundamental interplay between theory and application in linear algebra
- be able to solve problems by means of linear algebra
- Capacities of Effective Practice—including creativity and imagination, problem solving, self-directedness, resilience and persistence, and the abilities to collaborate with others across differences and to manage complexity and uncertainty. Students will
- apply their knowledge toward solving real problems
The text and class discussion will introduce the concepts, methods, applications, and logical arguments; students will practice them and solve problems on daily assignments, and they will be tested on quizzes, midterms, and the final.
- Textbook.
Linear Algebra, Concepts and Methods, by Martin Anthony and Michele Harvey. Cambridge University Press, 2012. ISBN 978-0-521-27948-2.
You may find that used books are less expensive than new ones.
Syllabus
We won’t cover all of the topics listed below at the same depth. Some topics are fundamental and we’ll cover them in detail; others indicate further directions of study in linear algebra and we’ll treat them as surveys. Besides those topics listed below, we will discuss some applications of linear algebra to other parts of mathematics and statistics and to physical and social sciences.
Matrices and vectors
Matrices.
Matrix addition and scalar multiplication.
Matrix multiplication. Matrix algebra. Matrix inverses.
Powers of a matrix. The transpose and symmetric matrices.
Vectors: their addition, subtraction, and multiplication by scalars (i.e. real numbers). Graphical interpretation of these vector operations
Developing geometric insight.
Inner products and norms in Rn: inner products of vectors (also called dot products), norm of a vector (also called length), unit vectors.
Applications of inner products in Rn: lines,
planes in R3, and
lines and hyperplanes in Rn.
Systems of linear equations
Matrix inversion and determinants
Matrix inverses.
Elementary matrices.
Introduction to determinants, 2x2 and 3x3 determinants,
areas of triangles and parallelograms in the plane, volumes of parallelepipeds, Jacobians
Characterizing properties and constructions of determinants,
cofactors, diagonal and triangular matrices.
More properties of determinants, an algorithm for evaluating
determinants, determinants of products, inverses, and transposes, Cramer’s rule.
Permutations and determinants.
Cross products.
Rank, range and linear equations
The rank of a matrix.
Rank and systems of linear equations.
Range.
Vector spaces
Linear independence, bases and dimension
Linear transformations and change of basis
Diagonalisation, eigenvalues and eigenvectors
Applications of diagonalisation
Inner products and orthogonality
Orthogonal diagonalisation and its applications
Orthogonal diagonalisation of symmetric matrices.
Quadratic forms.
Direct sums and projections
The direct sum of two subspaces.
Orthogonal complements.
Projections.
Characterizing projections and orthogonal projections.
Orthogonal projection onto the range of a matrix. Minimizing the distance to a subspace.
Fitting functions to data: least squares approximation.
Complex matrices and vector spaces
Complex numbers.
Dave’s Short Course on Complex Numbers.
Complex vector spaces.
Complex matrices.
Complex inner product spaces.
Hermitian conjugates.
Unitary diagonalisation and normal matrices.
Spectral decomposition.
Class notes, quizzes, tests, homework assignments
To be filled in as the course progresses.
- Some Assignments
- Exercises 1.1 through 1.7 page 53, and problems 1.1 through 1.6 page 55.
- Problems 1.8 through 1.14 page 57.
- Problems 2.1 through 2.8 page 86.
- Exercises 3.1 through 3.8, 3.11 page 125. (Note these are the exercises, not the problems.)
- Exercises 4.1 through 4.6, page 144.
- Problems 5.1 through 5.7, page 170.
- Various problems from chapters 6 and 7. (different assignments in the different class sections)
- Writing Proofs: structures of theorems and proofs,
synthetic and analytic proofs, logical symbols, and well-written proofs
- A little bit about sets
- Notes on Matlab
- The replacement theorem and dimension
- Where dimension doesn’t work
- Isomorphisms
- Algebra of linear transformations and matrices
- Kernel, image, nullity, and rank
- Rank and nullity of matrices
- Quaternions
Some old linear algebra tests
Web pages for related courses
Pages on the web that you may find interesting
This page is located on the web at http://aleph0.clarku.edu/~ma130/
William Rowan Hamilton (1805–1865)
