# Introductory_maths

A repository of tutorials to revise mathematical concepts required for statistics and machine learning
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# 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

Topics Tutorials
🔢 Introduction to numbers (Updated)
🔢 Introduction to algebra
🌗 Introduction to set theory (Updated)
:compass: Introduction to trigonometry
🍪 Introduction to summations
🍪 Introduction to combinatorics (Updated)
🔢 Introduction to functions
🎢 Introduction to derivatives
🎢 Introduction to integration
🎢 Differential equations
🎢 Multivariable functions
🎢 Differentiation of multivariable functions
🔢 Exponents and logarithms
🔢 Logarithms and information theory
🃏 Introduction to probability theory
🃏 Conditional probability
🃏 Bayes theorem
:compass: Introduction to distance metrics
:compass: Cosine similarity applications
:chopsticks: Introduction to linear systems
:chopsticks: Introduction to vectors
:chopsticks: Vector norms and embeddings
🏬 Introduction to matrices
:chopsticks: Linear transformations
:chopsticks: Applications of eigenvalues and eigenvectors

## 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 might feel archaic but they 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.

• 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

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