20933 - MATHEMATICS FOR AI - PREPARATORY COURSE

No background is required, other than basic mathematical knowledge.

This preparatory course introduces the basis of linear algebra and probability theory. In the first part of the course, we will cover some basic topics of linear algebra, including vectors, matrices, linear systems, vector spaces, linear maps, eigenvalues and eigenvectors, the spectral theorem, and the singular value decomposition. In the second part of the course, we will cover basic topics of probability theory, introducing discrete and continuous random variables, expectation, variance, Markov's Inequality and Chebyshev's Inequality.

Lecture 1 (28/08/23):

• Complex Numbers
• Vectors and Matrices
• Linear Systems
• Gaussian Elimination
   

Lecture 2 (29/08/23):

• Linear Combination of Vectors
• Vector Spaces
• Basis and Dimension of a Vector Space
   

Lecture 3 (30/08/23):

• Matrix Multiplication, Rank of a Matrix, Inverse Matrix, Trace of a Matrix
• Linear Maps and their Matrix Representation
• Kernel and Image of a Linear Map and the Rank-Nullity Theorem
• Injective and Surjective Linear Maps
   

Lecture 4 (31/08/23):

• Invertible Linear Maps and Isomorphism
• Computing an Inverse Matrix with the Gaussian Elimination
• The Rank of a Matrix (equivalent definitions)
• Determinant, Computing the Determinant with the Gaussian Elimination
• Norms and Inner Products
• Eigenvalues and Eigenvectors
   

Lecture 5 (01/09/23):

• Change of Basis
• Diagonalize a Matrix
• Spectral Theorem
• Positive Definite and Semidefinite Matrices
• Singular Value Decomposition
   

Lecture 6 (04/09/23):

I forgot to record the first part of Lecture 6. You can find a scan of my notes attached.

• Experiments, Probability, Events, Probability in Experiments with equally likely outcomes
• Permutations, Sampling with Replacement, Sampling without Replacement
• Binomial Coefficient, Multinomial Coefficient
• Probability Space, Axioms of Probability
• Conditional Probability and Independence of Events
• Bayes' Theorem and Law of Total Probability

  Lecture 7 (05/09/23):

• Discrete Random Variables
• Expectation, Linearity of Expectation
• Jensen's Inequality
• Variance and Standard Deviation
• Independent Random Variables
• Examples of Discrete Random Variables: Uniform, Bernoulli, Binomial, Poisson, Geometric
• Conditional Expectation
• Markov Inequality
• Covariance and properties of Covariance and Variance of two independent random variables
• Chebychev's Inequality
• Continuous Random Variables
• Examples of Continuous Random Variables: Uniform, Exponential

At the end of the course, the student will have basic knowledge of linear algebra and probability theory.
In particular, the linear algebra part of the course covers the following topics: vectors, vector spaces, matrices, linear maps, eigenvalues and eigenvectors, spectral theorem, and singular value decomposition. The probability part of the course covers the following topics: probability spaces, random variables, Markov Inequality and Chebychef inequality.

By the end of the course, students will know how to understand and solve basic exercises in linear algebra and probability theory. 

• Face-to-face lectures
• Online lectures
• Exercises (exercises, database, software etc.)

Classes are taken in person with the possibility of being taken online. In addition, all lectures are recorded.

The course has no exams.

Suggested textbooks:

•  Sheldon Axler, Linear Algebra Done Right
•  Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong, Mathematics for Machine Learning
•  Gilbert Strang, Introduction to Linear Algebra
•  Fabrizio Iozzi, Lecture Notes
• Sheldon Ross, A First Course in Probability
• Michael Mitzenmacher, Eli Upfal, Probability and Computing