MATH UN2015 – Linear Algebra and Probability
MATH UN2015 combines core topics in linear algebra with probability and selected applications. Linear algebra topics include systems of equations, matrices, vector spaces, bases, dimension, linear transformations, determinants, eigenvalues, and eigenvectors. Probability topics include conditional probability, random variables, probability distributions, limit theorems, and Markov chains.
The course also introduces applications such as least squares, regression, dimensionality reduction, network models, and dynamical systems. A project component explores applications in the natural and social sciences. No prior programming experience or background in these fields is required.
MATH UN2015 or MATH UN2010?
MATH UN2015 – Linear Algebra and Probability emphasizes applications and practical problem solving. It is well suited for students in engineering, technology, the natural and life sciences, the social sciences, economics, and data science.
MATH UN2010 – Linear Algebra places greater emphasis on theoretical foundations, abstract reasoning, and proofs. Mathematics majors and joint mathematics majors must take MATH UN2010 – Linear Algebra.
Students may not receive credit for both MATH UN2010 and MATH UN2015.
Prerequisite
MATH UN1101 – Calculus I, or the equivalent, is strongly recommended.
Textbooks
The following textbooks are optional but recommended:
- Deisenroth, M. P., Faisal, A. A., and Ong, C. S. (2020). Mathematics for Machine Learning. Cambridge University Press.*
- Bretscher, O. (2019). Linear Algebra with Applications, 5th ed. Pearson Education. See the publisher’s page.
- Barron, E. N., and Del Greco, J. G. (2020). Probability and Statistics for STEM: A Course in One Semester. Springer International Publishing.*
- Unpingco, J. (2020). Python for Probability, Statistics, and Machine Learning, 2nd ed. Springer Nature.* Code is available on GitHub.
Note: Materials marked with an asterisk are available electronically at no cost to Columbia and Barnard students.
Topics
Linear Algebra
- Systems of linear equations, matrices, vectors, and Gaussian elimination
- Vector spaces and subspaces
- Linear independence, bases, and dimension
- Linear transformations and matrix representations
- Image and kernel of a linear transformation
- Inner products, lengths, distances, angles, and orthogonality
- Orthonormal bases, orthogonal complements, and orthogonal projections
- Determinants and trace
- Eigenvalues and eigenvectors
- Spectral theorem
- Singular value decomposition and rank-\(k\) approximations
Probability
- Basic probability and conditional probability
- Bayes’ theorem
- Random variables and probability distributions
- Summary statistics and moment-generating functions
- Joint distributions and independence
- Covariance and correlation
- Central limit theorems
- Exponential families and conjugate distributions
- Change of variables and inverse-transform methods
Applications
- Markov chains
- Network flow and economic models
- Differential-equation models
- Least squares and data fitting
- Linear regression
- Principal component analysis
- Dimensionality reduction
Print this page