But what are Matrices, really? | Linear Algebra Explained

Summary

In this video, the presenter embarks on a 30-day challenge, delving into mathematics with a focus on linear algebra. They explore the concept of matrices, explaining them as multi-dimensional linear functions that take in vectors and output vectors. The video also highlights the historical significance of De Morgan's work in calculus and the evolution of mathematical notation.

Takeaways

• 📚 The video is part of a 30-day challenge focused on mathematics, particularly linear algebra.
• 🎓 The presenter is self-educating in high-level mathematics after finding university education insufficient for their needs.
• 📖 The script mentions an old book by De Morgan from 1842, which the presenter found valuable for understanding historical mathematical perspectives.
• 🕵️‍♂️ The presenter is on a quest to master high-level mathematics through self-education and documenting the journey for others to follow.
• 🎨 The video attempts to provide a heuristic for understanding the concept of matrices by creating a fictitious history that logically leads to their invention.
• 🔍 The presenter struggled to understand the concept of matrices beyond their manipulation rules and seeks to clarify their true meaning.
• 📈 The script explains the evolution from single-variable functions to multi-dimensional functions, leading to the concept of matrices.
• 🧠 The idea of matrices is introduced as a way to represent multi-dimensional inputs and outputs for functions, specifically in a linear context.
• 📉 The presenter emphasizes that matrices should be thought of as functions that take in vectors and produce vectors, operating in a linear manner.
• 📚 The study of linear algebra is described as the study of these linear, multi-dimensional functions, which includes concepts like eigenvalues, eigenvectors, and determinants.
• 🎉 The video concludes with the presenter's intention to do a series on linear algebra to further explore and explain the concepts introduced.

Q & A

What is the main focus of the video script?

The main focus of the video script is to discuss the concept of matrices and linear algebra, particularly from a historical and conceptual perspective. The script also mentions the author's personal journey in self-education and understanding higher-level mathematics.

Why did the author mention De Morgan and his book from 1842?

The author mentioned De Morgan and his book from 1842 to highlight the historical context and the evolution of mathematical concepts. De Morgan's work is significant in the field of calculus, and the author found an old copy of his book, which inspired the discussion.

What is the significance of De Morgan's law in the context of this script?

De Morgan's law is a fundamental concept in logic and set theory, but in this script, it serves as a connection to the historical roots of mathematical thought. The author uses it as a starting point to delve into the deeper understanding of matrices and linear algebra.

What is the author's approach to learning mathematics?

The author's approach to learning mathematics is self-education, using various resources and documenting the journey. The author has left traditional education methods, like university, to venture out on their own quest for knowledge.

Why does the author emphasize the idea of matrices as functions?

The author emphasizes the idea of matrices as functions to provide a conceptual framework that makes the abstract concept of matrices more understandable. By viewing matrices as functions that take in vectors and output vectors, the author aims to simplify the understanding of linear algebra.

What is the author's heuristic for learning mathematics?

The author's heuristic for learning mathematics involves creating a fictitious yet logically based history of a mathematical concept. This approach helps in making the learning process more relatable and easier to understand when compared to the actual historical development of the concept.

What is the role of linear equations in the discussion about matrices?

Linear equations play a crucial role in the discussion about matrices as they define the type of functions that matrices represent. The author explains that matrices are used to represent linear functions that take in vectors and output vectors in a multi-dimensional space.

How does the author describe the evolution of functions from single-variable to multi-dimensional?

The author describes the evolution of functions by starting with single-variable functions, then moving to two-variable functions, and finally to n-dimensional functions. This progression helps in understanding how functions can handle multiple inputs and outputs, leading to the concept of matrices.

What is the significance of the handwritten quote by Lord Kelvin in the script?

The handwritten quote by Lord Kelvin highlights the historical value and the impact of De Morgan's work on calculus. It also underscores the idea that De Morgan's book was highly valued but became less so due to its complexity, which was not suitable for examination purposes.

What are the main components of a matrix according to the author's explanation?

According to the author's explanation, the main components of a matrix are the coefficients that multiply the variables in a linear function. These coefficients determine how the input vectors are transformed into output vectors in a linear algebraic operation.