0. Introduction
Let's embark on an exciting journey of mastering Linear Algebra. We will start by exploring the concept of weighted sums of numbers and progress all the way to applying the powerful technique of Singular Value Decomposition. This e-book presents a series of interactive visual essays that incorporate code snippets, illustrations, animations, and LaTeX to effectively convey concepts in a highly intuitive manner.
A roadmap of this series
0. What will you be able to do after finishing this series?
0.1. Supercharge your problem solving skills
Linear Algebra provides a powerful set of problem-solving tools. By mastering it and integrating it with programming, you will gain invaluable problem-solving skills that can be applied in a multitude of real-world scenarios. Some applications of linear algebra in various fields are shown below:
Machine learning & data science
Linear regression, support vector machines, automatic differentiation
Computer graphics
Bezier and cubic curves, projections, intersections, linear transformations
Economics
Linear programming, econometrics, input-output models
Robotics
Kinematics, inverse kinematics, trajectory planning, homogeneous transformations
Data analytics
Principal component analysis, dimension reduction, data visualization
Digital signal processing
Filtering, convolution, Fourier transforms, spectral analysis
Physics
Classical & quantum mechanics, electromagnetism, statistical mechanics
Mathematics
Jacobians & Hessians, Covariance matrices, Markov processes, graph theory
0.2. Learn one of the most captivating topics in math
Linear Algebra is grounded in the fundamental concepts of scaling and adding vectors, which may seem straightforward at first glance. However, these concepts give rise to a rich and complex behavior that forms the core of this book. Exploring the properties of this emergent behavior unlocks a powerful toolkit for solving intricate and challenging problems.
1. A balanced approach
This series strikes a perfect balance between theory and application, with a focus on building a strong foundation in theoretical understanding and then applying it to practical scenarios. The content is designed to combine mathematical rigor with intuitive visual illustrations to aid comprehension.
2. Prerequisites
• You are expected to be familiar with basic geometry and trigonometry.
• If you understand simple code snippets in Python, that will help a lot. If you cannot, you can just ignore the code snippets and move on.
• These code snippets serve as an additional tool for explanation. There is no explanation on how the code is written. For example, the following code snippet elegantly illustrates the concept of matrix-matrix multiplication, conveying its meaning in a concise and effective manner:
def get_matrix_matrix_product(left_matrix, right_matrix):
return [
get_matrix_vector_product(left_matrix, column_vector)
for column_vector in right_matrix
]If you can understand this code, that's great. But if you can't, you can still rely on plain text and the good old latex.
3. How to go through this series?
The only thing worth mentioning here is the interactive illustrations like the one shown below. The icons alongside the visualizations are clickable. In the example below, clicking the button changes the matrix displayed.
Matrix: \(
\begin{bmatrix}
1 & 0 \\ 0 & 1
\end{bmatrix}
\)
4. Comparison with other resources
Some other great resources worth mentioning here are:
• MIT OCW Course by the legendary Prof. Gilbert Strang: This is without a doubt one of the best courses out there to learn linear algebra. And its totally free of cost.
• Linear Algebra Done Right by Sheldon Axler: This is a great textbook but it may be a bit overwhelming for an absolute beginner.
• Essence of Linear Algebra by 3blue1brown: It's a great starting point but it only provides a basic introduction to the topic.
• Video lectures by MathTheBeautiful: I think it is one of the under-rated gems on YouTube.
4.1. Why this series then?
• It covers more applied topics and problems like Barycentric co-ordinates, Bezier curves, inverse of transpose of a matrix etc.
• The digital medium allows using visual aids, animations and an interactive format to convey information in a more engaging & intuitive way.
• It is fully contained and comprehensive - by the end of this series, you will know enough linear algebra required to become a data scientist.
4.2. What's missing in this series?
• Inspired from Axler's approach, I did not mention determinants at all throughout this e-book.
• I also skipped certain topics like Gaussian elimination, echelon forms and Gram Schmidt orthogonalization. I may add these topics later in the future.
• It is missing some common matrices found in other areas of mathematics: Jacobians and Hessians in calculus, Laplacian matrix in Spectral Graph Theory, Fourier matrix etc. I do talk about covariance matrix in the last essay.
Let's get started now.