Math 19b, Linear Algebra, Probability and Statistics, Spring, 2011

Syllabus

Mathematics 19b: Linear Algebra, Probability, and Statistics for the Life Sciences

Harvard College/GSAS: 6144, Exam Group: 6

Spring 2010-2011 Oliver Knill (knill.harvard.edu) Monday/Wednesday/Friday, at 1 in Hall E. Weekly problem section to be arranged. This course features linear algebra focusing more on probability, statistics. It contains applications to life sciences, chemistry, and environmental life sciences. No a priori background in the life sciences is assumed.

The course covers essentially all of the linear algebra covered in Math21b or applied Math21b: matrices, eigenvalues, eigenvectors, determinants. It also teaches applications of linear algebra to probability, statistics and dynamical systems giving a background sufficient for higher level courses in statistics like Stat 111. The course teaches the subjects linear algebra, probability and statistics hand in hand. In the probability part we see standard models and techniques, and their uses including the central limit theorem, Markov chains, curve fitting, regression, and pattern analysis. A successful completion of this course will provide you with a level of sophistication in mathematics that will serve you well in your future science courses, and in your future scientific career.

Students who have seen some multivariable calculus can take Math 19b before Math 19a. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.

Textbook: Otto Bretscher's "Linear Algebra with Applications" as well as the course notes by Cliff Taubes "Probability, Statistics and Linear Algebra" which will be posted here.

Homework: homework is assigned each class and due on Wednesdays. Homework assignments submitted after their assigned due date will be accepted only at the discretion of the instructor. No more than 2 late homework assignments will be accepted per student over the course of the semester.

Exams: There will be a single midterm exam and a final exam. The midterm is a take home exam, given over from March 23-March 25. The final exam date will be announced later in the term.

Who should take this course: This course is recommended over Math 21b for those planning to concentrate in the life sciences, chemistry, or ESPP. It can be taken with Mathematics 21a. You should take this course if you are considering Math 21b or Applied Math 21b and are also thinking of concentrating in one of the life sciences. You should also take this course if you have taken Math 19a. Those coming from Math 19a will have seen already much of the differential equations that are taught in Math 21b. Note that you can take Math 19b without having taken Math 19a.

Prerequisits: are either Math 1b with a satisfactory grade, or a AB-BC score of at least 4, or scores of at least 20, 8, 4 on the respective three Harvard University Math Placement Tests.

Lectures
Week1: Probability theory:  Conditional probability
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Lecture 1                Chapter 1:  Data and linear algebra
Lecture 2                Chapter 2:  Probability spaces
Lecture 3                Chapter 3:  Conditional probability

Week 2: Linear algebra:  Linear equations
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Lecture 4                Chapter 1.1: From Bayes to linear equations
Lecture 5                Chapter 1.2: Gauss-Jordan elimination
Lecture 6                Chapter 1.3: Rank and the number of solutions

Week 3: Linear algebra: linear transformations
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Lecture 7                Chapter 2.1: Matrices and Linear transformations
Lecture 8                Chapter 2.3: Linear transformations
Lecture 9                Chapter 2.4: Matrix algebra

Week 4: Probability:      Random variables 
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Lecture 10               Chapter 4: Random variables
Lecture 11               Chapter 5: Independence 
Lecture 12               Chapter 6: Parameter estimation

Week 5: Linear algebra:    Basis of image and kernel
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Presidents day, no class
Lecture 13               Chapter 3.1: Image and kernel
Lecture 14               Chapter 3.2: Basis of linear spaces

Week 6: Linear Algebra   Coordinate changes, Orthogonality 
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Lecture 15               Chapter 3.3: Dimension
Lecture 16               Chapter 3.4: Coordinates
Lecture 17               Chapter 5.3: Orthogonal transformations

Week 7: Linear algebra:    Data fitting 
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Lecture 18                Chapter 5.1: Orthogonal projection
Lecture 19                Chapter 5.4: Regression and Least squares
Lecture 20                Chapter 5.4: More fitting problems

Spring break 
 
Week 8: Probability       Probability distributions
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Lecture 21                Review for the midterm      
Lecture 22                Chapter 11: Probability distributions
Lecture 23                Chapter 12: Chebyshev's theorem

Week 9: Linear algebra   Determinants
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Lecture 24                Chapter 6.1:  Determinants
Lecture 25                Chapter 6.2:  Determinants
Lecture 26                Chapter 6.3:  Geometry of determinants

Week 10: Linear algebra   Eigenvalues and Eigenvectors

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Lecture 27                Chapter 7.1   Eigenvalues
Lecture 28                Chapter 7.2   Eigenvectors
Lecture 29                Chapter 7.3   Eigenvectors   

Week 11: Probability:     Central limit theorem
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Lecture 30                Chapter 7.4   Diagonalization
Lecture 31                Chapter 13    Law of large numbers
Lecture 32                Chapter 13    Central limit theorem 

Week 12: Probability:     Markov processes
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Lecture 33                Chapter 16:  Markov processes
Lecture 34                Chapter 17:  Perron Frobenius
Lecture 35                Chapter 8.1  Spectral Theorem

Week 13: Linear algebra:   Symmetric matrices
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Lecture 36                              Review I
Lecture 37                              Review II