An Interactive Journey from Equations to Structure
Interactive diagrams. Step-by-step algorithms. Conceptual quizzes. Everything you need to deeply understand linear algebra — not just compute it.
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Chapter 1
Geometry, Structure, and Solution Methods
Discover how systems of equations encode geometry, develop powerful elimination algorithms, and reveal the deep structure of solution sets.
Chapter 2
Span, Independence, and the Language of Linear Algebra
Master the language of vectors: how to add, scale, combine, and span with them — and discover the deep notions of linear independence, subspace, and basis.
Chapter 3
Rank, Nullity, and the Structure of Transformations
Understand matrices as linear transformations, master the column space and null space, and discover the Rank-Nullity theorem that ties everything together.
Chapter 4
Composition, Products, and LU Decomposition
Master matrix multiplication as composition of transformations, understand invertibility through 10 equivalent conditions, and learn the LU decomposition algorithm.
Chapter 5
Rows, Columns, and a Surprising Equality
Discover the transpose operation and explore the deep symmetry between row space and column space through the Row Rank Theorem — one of the most elegant results in linear algebra.
Chapter 6
Measuring Angles, Length, and Finding the Closest Vector
Build a complete theory of measurement in linear algebra — dot products, lengths, angles, and orthogonality — then use orthogonal projection to solve the best approximation problem.
Chapter 7
Directions That Only Scale
Discover the special directions a matrix never rotates — only stretches or shrinks. Eigenvalues and eigenvectors reveal the hidden structure of a linear transformation and unlock powerful tools in data science, physics, and differential equations.
Chapter 8
When Matrices Meet Geometry
Explore how symmetric matrices and their eigenvalues determine the shape of quadratic functions — from bowls to saddles. Build toward positive definiteness, the spectral theorem, and the geometry behind least squares.
Chapter 9
The Universal Matrix Factorization
Master the SVD — the most powerful factorization in linear algebra. Understand how any matrix decomposes into rotations and scalings, how it generalizes the spectral theorem, and how it underpins PCA, low-rank approximation, and multidimensional scaling.
Chapter 10
The Calculus of Finding Minima
Build from scratch the calculus tools needed for optimization: limits, continuity, derivatives, and differentiability. Understand local linear approximation and the hierarchy of function classes C⁰, D¹, and C¹.
Chapter 11
Extending Differentiation to Many Variables
How do we differentiate functions of multiple variables? This chapter builds the toolkit: partial derivatives, local linear approximation, differentiability, and the Jacobian matrix — the multivariable generalization of the derivative.
Chapter 12
Finding Minima Using First- and Second-Order Information
Building on partial derivatives and the Jacobian, this chapter introduces the gradient vector, directional derivatives, gradient descent, critical point theory, and the practical methods for finding global minima of functions on subsets of Rⁿ.
Chapter 13
Functions from Rⁿ to Rᵐ and Their Derivatives
Extending differentiation to functions with vector outputs. This chapter covers the Jacobian for vector-valued functions, local linear approximation in the vector setting, trajectories, and the geometry of vector-valued maps.
Chapter 14
Differentiating Compositions of Functions
The chain rule is the master tool of calculus, enabling the differentiation of composed functions. This chapter derives the multivariate chain rule via local linear approximation, explores matrix-form justification, and covers the key special cases: one underlying variable, two underlying variables, and non-canonical input/output types.
Chapter 15
Computing Gradients Through Computational Circuits
Backpropagation is the algorithm that makes training neural networks tractable. By viewing computation as a circuit, we can compute all partial derivatives in two efficient passes — one forward, one backward.
Chapter 16
Curvature, the Hessian, and Globally Well-Behaved Functions
Second derivatives capture curvature — how fast a function bends. The Hessian matrix organizes all second partials, powers the second derivative test, and unlocks the powerful theory of convex functions where local minima are guaranteed to be global.
Chapter 17
Optimizing Under Equality Constraints
When optimization is restricted to a surface defined by equality constraints, gradient descent alone fails. Lagrange multipliers provide the systematic tool, revealing that constrained optima occur where gradients of the objective and constraints align — and even proving that symmetric matrices have eigenvalues.