Introductory mathematics in R and Python

This repository contains tutorials on the introductory mathematical concepts required for studying statistics and machine learning. Code to solve mathematical problems is written in R, Python and Julia.

Tutorials

General concepts

• Numbers (Updated)
• Exponents and logarithms
• Logarithms and information theory
• Geometry
• Cosine similarity applications
• Distance metrics
• Summations
• Functions
• Algebra (Updated)

Linear algebra

• Linear systems
• Introduction to vectors
• Vector norms and embeddings
• Introduction to matrices
• Linear transformations
• Applications of eigenvalues and eigenvectors

Calculus

• Derivatives
• Integration
• Differential equations
• Multivariable functions
• Differentiation of multivariable functions

Statistical mathematics

• Set theory
• Combinatorics
• Probability theory
• Conditional probability
• Bayes theorem

Contributors

• Erika Duan
• Chuanxin Liu

Project setup

This project was created using the following setup:

• R package dependencies are managed using renv for R version 4.1.2 (2021-11-01).
• Python virtual environment and package dependencies are managed using poetry for Python 3.9.6. A local version of Python 3.9.6 was installed and activated using pyenv local 3.9.6 via the terminal.
• The Julia version used is julia version 1.7.3.

Guide to writing mathematical proofs

Writing mathematical proofs are a great way to help you reason why mathematical concepts should behave consistently (and not just because your textbook says so). There are multiple approaches to proving a mathematical statement or concept. Sadly, there is no magical rule to selecting the correct method for each scenario - mathematicians often have to try multiple approaches before they find the right one.

Direct proof

• Occurs when you need to prove that A and B are equivalent.
• Start by assuming A is true.
• Construct a definition statement for A (use a fixed but arbitary example of A).
• Extend and simplify mathematical definitions derived from A to reach B.
• When you are asked if A and only A is true, then B is true, first suppose A to reach B. Then suppose B to reach A.

Induction proof

• Occurs when you need to prove that something is true for all cases.
• Start by proving the base case when $n = 1$.
• Assume that the case is also true for some integer $k$.
• Prove that the case for $k + 1$ also holds i.e. prove the next incremental step up a ladder stretching to infinity.

Uniqueness proof

• Occurs when you need to prove that a solution is unique.
• Show that there is one solution first.
• Show that there is a second solution and that the first and second solutions must be equal.

Proof by contradiction

• Start by assuming that the incorrect state is true i.e. that eigenvectors are linearly dependent.
• Prove that the assumption does not hold and contradicts itself.
• Therefore prove that the reverse state is actually true.

References

• A guide to linear algebra for applied machine learning by Pablo Caceres
• The Mathematics for Machine Learning textbook by Marc Peter Deisenroth, A Aldo Faisal and Cheng Soon Ong - Cambridge University Press
• The Probability for Data Science textbook by Stanley H Chan - Michigan Publishing
• The Probabilistic modelling tutorials by Michael Betancourt - GitHub
• Writing mathematical operations in LaTex/R - Wikibooks
• Introduction to university mathematics [YouTube lecture series ]https://www.youtube.com/playlist?list=PL4d5ZtfQonW1xKVEtYJd1iu9m52ATG7SV by the Department of Mathematics - Oxford University.