Learn Linear Algebra with This 20-Hour Course and Free Textbook

Linear algebra is the bedrock of modern mathematics and a critical tool for science, engineering, and computing. It‘s a pillar of any serious technical education, and a requirement for those hoping to make an impact in cutting-edge fields like AI, robotics, computer graphics, data science, and scientific computing.

In the words of renowned mathematician Gilbert Strang, "Linear algebra is a fantastic subject to learn. It‘s a combination of four or five central ideas, and all you have to do is understand them and how they interact. Then you can answer almost any question."

While linear algebra has a reputation for being abstract and difficult, it‘s surprisingly learnable with the right approach. And now, thanks to a free course from Professor Jim Hefferon of St. Michael‘s College, anyone with a basic background in algebra can dive in and start mastering this essential subject.

A 20-Hour Video Course and Free Textbook

Professor Hefferon‘s linear algebra course, available on the freeCodeCamp YouTube channel, provides a comprehensive introduction to undergraduate-level linear algebra. The course comprises about 20 hours of video lectures that follow along with Hefferon‘s free textbook, which you can download from his website.

The lectures cover all the standard topics in a first linear algebra course, including:

• Systems of linear equations and their solution by Gaussian elimination
• Vectors, linear independence, bases, and dimensions
• Linear transformations and matrix representations
• Determinants, eigenvalues, and eigenvectors
• Inner products and orthogonality
• Abstract vector spaces and linear maps

Hefferon‘s explanations are clear, concise, and example-driven. He takes care to motivate key concepts and connect them to concrete applications. The pacing is measured, with ample time devoted to working through problems and clarifying tricky ideas.

Alongside the lectures, it‘s crucial to read the textbook and work through the many practice problems it provides. Hefferon‘s book is extraordinarily well-crafted, with hundreds of examples, figures, and exercises. Every major concept is illustrated visually and computationally as well as algebraically.

Applications in Computer Science and Software Engineering

For students of computer science and software engineering, linear algebra is an invaluable tool. It provides the mathematical language for reasoning about high-dimensional spaces, modeling complex systems, and designing efficient algorithms. Here are just a few of the ways linear algebra powers modern computing:

• In computer graphics, linear algebra is used to manipulate 3D models, apply transformations, and project scenes onto 2D screens. Libraries like OpenGL and Direct3D are built on a foundation of vector and matrix operations.
• Machine learning algorithms like linear regression, principal component analysis, and support vector machines are all formulated in terms of linear algebra. Concepts like least squares optimization and singular value decomposition are central to training robust models.
• In cryptography, linear algebra is used to analyze and design encryption schemes. The hardness of problems like integer factorization and discrete logarithms, which underpin protocols like RSA, boils down to questions in linear algebra over finite fields.
• Quantum computing, an emerging paradigm that promises exponential speedups on certain problems, is essentially linear algebra over complex vector spaces. Quantum states are represented by vectors, and quantum gates are unitary linear transformations.

The demand for linear algebra expertise is surging as these fields continue to grow. According to a recent report from Burning Glass Technologies, linear algebra is now one of the most sought-after skills for data science and analytics jobs, with demand growing 740% since 2015.

A Few Key Concepts

To give you a flavor of the power and elegance of linear algebra, let‘s briefly explore a few key concepts from Professor Hefferon‘s course.

Eigenvectors and Eigenvalues

Eigenvectors and eigenvalues are perhaps the most important concepts in applied linear algebra. Intuitively, an eigenvector of a linear transformation is a vector that remains on its own span after being transformed. Each eigenvector has an associated eigenvalue that scales the vector when transformed.

Formally, if $A$ is an $n \times n$ matrix, a nonzero vector $\mathbf{x}$ is an eigenvector of $A$ if there exists a scalar $\lambda$ such that:

$$A\mathbf{x} = \lambda\mathbf{x}$$

The scalar $\lambda$ is the eigenvalue corresponding to $\mathbf{x}$.

Eigenvectors and eigenvalues have numerous applications, from analyzing the stability of differential equations to uncovering hidden structure in data via principal component analysis. They also allow us to classify and simplify linear transformations by "diagonalizing" their matrix representations.

Singular Value Decomposition

The singular value decomposition (SVD) is a powerful tool for understanding the geometry of linear transformations. It decomposes any matrix $A$ into a product of three simpler matrices:

$$A = U\Sigma V^T$$

Here $U$ and $V$ are orthogonal matrices (representing rotations or reflections), and $\Sigma$ is a diagonal matrix of singular values (representing stretches or shrinks).

Geometrically, the SVD shows that every linear transformation can be broken down into a composition of rotations and stretches along perpendicular axes. It also identifies the "principal directions" of the transformation – the directions in which it stretches or shrinks vectors the most.

The SVD has countless applications, from data compression and noise reduction to solving systems of equations and analyzing the sensitivity of numerical algorithms. It‘s a Swiss Army knife for working with matrices and a cornerstone of numerical linear algebra.

Tips for Learning Linear Algebra

As a full-stack developer who‘s had to learn and apply linear algebra on the job, here are a few tips I‘d offer for getting the most out of your studies:

1. Code it up. Implementing linear algebra algorithms from scratch is the best way to truly understand them. Write functions for Gaussian elimination, matrix multiplication, eigenvalue problems, etc. Test them on simple examples and verify that they match the theory.

2. Use NumPy (or your favorite numerical computing library). While coding from scratch is instructive, you‘ll want to use optimized routines for real-world problems. Learn a library like NumPy and practice using it to solve LA problems efficiently.

3. Visualize everything. Linear algebra is inherently geometric, so build visual intuition wherever possible. Sketch transformations of the plane, plot vectors in 3D, visualize matrices as grids of numbers. Use tools like Matplotlib and GeoGebra to explore concepts interactively.

4. Connect linear algebra to your domain. Find examples of linear algebra at work in your field and strive to understand them deeply. Read papers, work through case studies, and talk to colleagues who use LA in their work.

5. Teach others. One of the best ways to solidify your understanding is to explain it to someone else. Tutor a classmate, write blog posts about what you‘re learning, or give a talk at a local meetup. Teaching will quickly reveal gaps in your knowledge and force you to clarify your thinking.

Conclusion

Linear algebra is a phenomenally rich and useful branch of mathematics, one that underpins much of modern science and technology. With Professor Hefferon‘s free course and textbook, you have everything you need to start exploring this fascinating subject and unlocking its applications.

But the true reward of learning linear algebra is not just its utility, but the elegance and power of its central ideas. As you study, take time to marvel at the deep connections it reveals and the surprising insights it offers. Let its beauty inspire you, just as it has inspired generations of mathematicians and scientists.

In the words of mathematician Paul Halmos, "Linear algebra is not just a collection of tricks, or a handful of techniques; it‘s a way of looking at mathematics and the world." Embrace that way of looking as you learn, and there‘s no telling what discoveries and innovations you‘ll be empowered to make.