Day 4 of 30 Day Challenge

Introduction

• Focus: Linear Algebra and Mathematics
• Self-education journey in high-level mathematics
• Exploring old texts for insights (De Morgan's book from 1842)

De Morgan's Book

• Encountered an old calculus book by De Morgan, noted for its unique spelling (e.g., "shew" instead of "show")
• Book published in 1842, handwritten quotes by Lord Kelvin
• Kelvin's quote: De Morgan's calculus work is highly valued but not suitable for examinations

Understanding Matrices

• Presentation shares a fictitious history of how matrices were conceived
• Importance of understanding the logic behind mathematical concepts

Functions as a Starting Point

• A function is a machine that takes one input and produces one output (e.g., f(x) = y)
• Evolution of functions: introducing multiple inputs and outputs

Multi-dimensional Functions

• Transition from single output to multiple outputs (e.g., two inputs producing two outputs)
• Introduced a new notation for multi-dimensional outputs
• Example: Functions defined with multiple variables (e.g., f(x, y) = (z, w))

Concept of Matrices

• Matrices represent multi-dimensional functions
• A matrix takes a vector as input and produces a vector as output
• They are a way to handle multi-dimensional linear outputs
• Key: Matrices are functions that operate linearly

Linear Equations

• Definition: A linear equation only involves variables multiplied by coefficients
• Non-linear examples: y = sin(x), y = e^x
• Linear must not have constant terms added to variables

Matrices as Functions

• Matrices can have various dimensions, e.g., taking a 2D vector to a 4D vector
• The study of linear algebra revolves around these functions
• Consider matrices as a function in multidimensional space

Conclusion

• Key takeaway: See matrices as functions to clarify understanding
• Next steps: Future discussions on linear algebra concepts
• Intention to continue the series and improve clarity in future presentations

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• Students are encouraged to explore the concepts further and engage with the material.