# 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 --------------------------------------------------- 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 ---------------------------------------------- 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 ---------------------------------------------------- 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 -------------------------------------------- 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 ------------------------------------------- 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 --------------------------------------------- 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 --------------------------------------------- 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 -------------------------------------------- Lecture 21 Review for the midterm Lecture 22 Chapter 11: Probability distributions Lecture 23 Chapter 12: Chebyshev's theorem Week 9: Linear algebra Determinants --------------------------------------------- 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 ---------------------------------------------- Lecture 27 Chapter 7.1 Eigenvalues Lecture 28 Chapter 7.2 Eigenvectors Lecture 29 Chapter 7.3 Eigenvectors Week 11: Probability: Central limit theorem -------------------------------------------- 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 -------------------------------------------- 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 --------------------------------------------- Lecture 36 Review I Lecture 37 Review II