Linear Algebra for Programmers, Vectors to SVD (Python & C#)

Learn how to compute and interpret the dot product of two vectors, both numerically and geometrically.

Discover how it relates to vector length, angles, and real-world applications such as 3D graphics and AI similarity measures.

Introductie of matrix multiplication and its key properties—commutativity, associativity, and inverses—and shows how these ideas lead to solving systems of linear equations in modern computing.

In this lesson, you’ll learn how transformation matrices change vectors and shapes in 2D and 3D. You’ll apply scaling, rotation, reflection, shear, and translation, and see how matrix multiplication performs these transformations in both geometry and data.

In this lesson, you’ll explore linear dependence and independence of vectors, and how they determine whether systems like Ax = b have unique solutions. You’ll learn about rank, span, and null space, and see why these concepts matter for understanding data, features, and motion in AI, data science, and robotics.

In this lesson, you’ll learn how to handle underdetermined and overdetermined systems of equations. You’ll use projections and the least squares method to find the best approximate solution, explore the normal equations, and see how these ideas power applications in AI, robotics, and computer graphics.

In this lesson, you’ll learn how matrix factorizations simplify solving linear systems. You’ll explore LU, QR, and Cholesky decompositions, see how they replace inversion with efficient step-by-step solves, and understand when each method — or iterative solvers — is best suited for real-world problems.

In this lesson, you’ll learn what eigenvalues and eigenvectors are, how to compute them, and how they relate to the decomposition A = VΛV⁻¹. You’ll see how they describe transformations, stability, and dimensionality reduction, and explore their key roles in PCA, PageRank, simulations, and computer