Course Overview

This course aims to provide an intuitive and comprehensive understanding of various mathematical topics that are vital for anyone seeking a deep understanding in fields like physics, computer science, and engineering. This is not your normal course as it is designed to be as humanely fast as possible to intake all the mathematical concepts you need as fast as possible. The

Course Overview

Welcome to our Intuitive Mathematics Course! This unique learning journey will take you through a wide array of topics across mathematics, physics, engineering, and programming. We'll start with the basics and build up to the more complex stuff.

What Will We Cover?

• Mathematics: Key topics we'll explore include Calculus, Linear Algebra, Abstract Algebra, Functional Analysis, Complex Analysis, Fourier Analysis, Probability and Statistics, Differential Geometry, Numerical Analysis, Topology, Lie Algebras, Representation Theory, and Graph Theory.
• Physics and Engineering: We'll put our math skills to work in understanding Classical Mechanics, Thermodynamics, Statistical Mechanics, Electrodynamics, Special and General Relativity, Classical Field Theory, Particle Physics, Nuclear Physics, Condensed Matter Physics, Astrophysics, and Cosmology.
• Programming Concepts: Along the way, we'll also dive into Programming Fundamentals, Object-Oriented Programming, Functional Programming, Quantum Programming, Computer Architecture, Parallel and Distributed Computing, Software Engineering Practices, Cryptography, and Machine Learning.

How Will We Learn?

1. Logical Sequence:Order is based off my own intuition and learning journey through these topics. The most foundational and important concepts and connections are covered first with an emphasis on understanding the importance of topics and relational importance with regard to other topics.
2. GPT-4 Technology: We'll use the power of GPT-4 to check your understanding and help you explain concepts in your own words.
3. Repetition & Reinforcement: After encoding these foundational concepts, a program of spaced interleaved repetition will be given to the user to effectively retrieve all relevant information.

Jump in and get started on your fast-track learning journey in mathematics, physics, computer science, and engineering!

Learning Outcomes

• Develop an intuitive understanding of complex mathematical concepts.
• Learn to apply mathematical concepts in various fields.
• Improve problem-solving abilities using mathematical thinking.

Introduction

We are going to start with algebra. The most fundamental concepts of algebra are:

• Vector spaces
• Linear transformations & Matrices
• Orthogonal and unitary operators
• Eigenvalues and eigenvectors

Vector Spaces

Vector spaces provide a way for us to represent many physical phenomena such as force, velocity, and position etc. The vector axioms are rules mathematicians derived from experimental observation. They provide a description of how velocities can be added to each other and scaled, of position and movement etc. It happens that the rules that govern velocities and position and movement are all the same.

These are the vector axioms. They are the way because they are useful. We can remember all the axioms later. Remember, this course is focused on efficiency. What we need to remember is that vectors can be scaled and added to each other within the same vector space as much as we like and the resultant vector will always be in the same vector space.

Linear Transformations

Linear transformations are basically something that will take one vector, and output another vector. The question is, if we linearly transform every single vector within one vector space to another, will the set of all the transformed vectors form another vector space? Well, not necessarily (if it's not linear).

If we make a transformation (G) that takes each vector v in V to the scalar 1, the set of {1} is obviously not a vector space. Why is it not? Well the first axiom of vector spaces is additivity. Every possible additive combination of vectors in the space is also a vector in the space. We could have 1+1 =2. Which isn’t in the set, therefore it is not a vector space.

Now, a linear transformation is a transformation that preserves scalar multiplication and additivity within the set of transformed vectors.