Linear Algebra

Bezier Curves

Parametric curves for computer graphics.

Matrix Multiplication

Visualize how 2x2 matrices transform space and combine through multiplication.

Dot Product & Projection

Visualizing vector projection and orthogonality.

Eigenvalues & Eigenvectors

Discovering the special directions that remain unchanged under linear transformations.

Linear Transformations

Visualizing 2D transformations via matrix operations.

Cross Product

Cross Product

Determinant & Area

Visualize how a 2x2 matrix transforms the area of a unit square.

Gram-Schmidt Process

Gram-Schmidt Process

Systems of Linear Equations

Visualizing the intersection and solution of multiple linear equations.

Vector Spaces

Vector Spaces

Change of Basis

Visualizing how vector coordinates transform between different coordinate systems.

LU Decomposition

LU Decomposition

Least Squares Approximation

Explore how least squares approximation fits a line to data points by minimizing the sum of squared residuals.

Markov Chains

Visualize state transitions, probability distributions, and steady states in stochastic systems.

Null Space & Column Space

Visualize the null space, column space, and rank-nullity theorem of a 2x2 transformation matrix.

QR Decomposition

Visualize the decomposition of a matrix into orthogonal and upper triangular components.

Affine Transformations

Visualize transformation consisting of a linear map followed by a translation.

Cramer's Rule

Visualize Cramer's Rule for solving a 2x2 system of linear equations.

Diagonalization

Visualize matrix diagonalization as changing to an eigenbasis to scale coordinates.

Jordan Normal Form

Visualize continuous phase portraits and canonical forms of 2x2 Jordan blocks.

Orthogonal Projections

Visualize orthogonal projections of vectors onto subspaces and compute projection matrices.

Positive Definite Matrices

Visualize positive definite matrices and their quadratic forms.

Power Iteration Method

Visualize the power iteration algorithm converging to the dominant eigenvector.

Rank & Nullity Theorem

Visualize how 3D transformations split space into image and kernel dimensions.

Spectral Theorem

Visualize how symmetric matrices are guaranteed to have real eigenvalues and orthogonal eigenvectors.

SVD Decomposition

Visualize Singular Value Decomposition as a sequence of rotation, scaling, and rotation transformations.

Tensor Operations

Visualize tensor contractions and transformations on multi-dimensional matrices.

Cholesky Decomposition

Visualize the factorization of a symmetric positive-definite matrix into a lower triangular matrix and its transpose.

Givens Rotation

Visualize how a Givens rotation zeroes out specific elements in a vector while preserving its norm.

Householder Reflection

Visualize how a Householder matrix reflects a vector across a hyperplane.

Inner Product Spaces

Visualize how different inner products change the concepts of length, angle, and orthogonality.

Least Norm Solution

Find the minimum energy solution to an underdetermined system of linear equations.

Matrix Exponential

Visualize the continuous evolution of systems using the matrix exponential.

Matrix Norms & Condition Number

Visualize how matrix norms measure size and condition number affects numerical stability.

Cayley-Hamilton Theorem

Visualize how every square matrix satisfies its own characteristic equation through vector transformations.

Iterative Solvers (Jacobi)

Visualize the convergence of the Jacobi iterative method for solving systems of linear equations.

Kronecker Product

Visualize how the Kronecker product combines two matrices to form a larger block matrix.

Moore-Penrose Pseudoinverse

Visualize how the Moore-Penrose pseudoinverse finds the least squares solution for linear systems.

Schur Decomposition

Visualize Schur decomposition as the orthogonal transformation of a matrix into upper triangular form.