Geometry of Linear Systems
Visualize the row and column pictures for 2D linear systems.
Solve SystemLinear Algebra
Matrix operations, geometric transformations, decompositions, and least squares.
This collection covers a broad range of topics in linear algebra, from foundational matrix operations to advanced decompositions and applications.
Foundations
- Row Echelon Form & RREF — step-by-step Gaussian elimination with pivot highlighting
- Elementary Matrices — visualize each row operation as a matrix multiplication
- Determinant & Trace — compute and interpret for square matrices
- Determinant Visualizer — see the determinant as oriented area or volume
- Matrix Inverse — find inverses via augmented row reduction
Transformations & Geometry
- 2D Transformations — rotation, reflection, scaling, shear, projection with animated objects
- Geometry of Linear Systems — row and column pictures for 2D systems
- Change of Basis — transition matrices and coordinate transforms
Decompositions & Advanced Topics
- Eigenvalues & Eigenvectors — compute and visualize for real matrices
- SVD — singular value decomposition with 2D geometric visualization
- PA = LU, QR, Cholesky, A = CR — standard matrix factorizations
- Least Squares — solve via normal equations, QR, or SVD
- Four Fundamental Subspaces — column space, null space, row space, left null space
Tool Descriptions
2D Visualizations
2D Matrix Transformation Visualizer
Visualize rotation, reflection, scaling, shear, projection, and translation with animated objects. See SVD decomposition and compose transformations.
▶ Animate2D SVD Visualization
Visualize the SVD of any 2x2 matrix as three geometric steps: rotate (VT), scale (Σ), rotate (U). See rank deficiency collapse objects to a line.
▶ Full TransformCoordinate Transform Visualizer
Animated visualization of change-of-basis for 2D vectors. See how coordinates transform between different bases with interactive examples.
▶ AnimateMatrix Basics
Matrix Product
Interactive computation and visualization of matrix multiplication as composition of linear transformations.
▶ ComputeDeterminant and Trace
Compute the determinant and trace of small matrices using elementary matrix operations.
ComputeSolving Linear Systems
Reduced Row Echelon Form Solver (RREF)
Step-by-step Gauss-Jordan elimination to compute the reduced row echelon form.
Calculate RREFMatrix Inverse
Compute the inverse of a square matrix via augmented row reduction.
Calculate InverseElementary Matrices
Explore how elementary row operations are represented by multiplication by elementary matrices.
Calculate REFRow Echelon Form Solver (REF)
Perform Gaussian elimination to compute the row echelon form.
Calculate REFBasis, Similarity & Orthogonality
Change of Coordinates (Transition Matrix)
Convert vector coordinates between bases using the transition matrix.
ComputeSimilarity Transforms
Find the matrix representation of a linear operator.
ComputeGram-Schmidt Orthogonalization
Apply the Gram-Schmidt process to obtain an orthogonal (or orthonormal) basis step-by-step.
CalculateMatrix Diagonalization
Diagonalize a matrix using its eigenvectors when possible.
Compute DiagonalizationCholesky Factorization
Compute the Cholesky decomposition for symmetric positive definite matrices.
ComputeFour Fundamental Subspaces
Compute bases for the column space, null space, row space, and left null space of a matrix.
ComputeFactorizations & General Solvers
A = CR Factorization
Compute the rank-revealing factorization of a matrix.
ComputeA = CMR Factorization
Compute the rank-revealing factorization \(A = CMR\), where \(M = W^{-1}\).
ComputeLinear Equation Solver
Solve a linear system with detailed step-by-step row reduction.
SolveMatrix Inverse or Pseudo-Inverse
Compute the inverse (if it exists) or the Moore-Penrose pseudo-inverse.
ComputePA = LU Factorization
Compute the LU decomposition with partial and scaled partial pivoting.
Compute LUQR Factorization
Compute the orthogonal-triangular decomposition.
ComputeLeast Squares, Eigenvalues & SVD
Least Squares Solver using QR Factorization
Solve the least squares problem using QR decomposition.
ComputeLeast Squares Solver using Normal Equations
Solve the least squares problem using the normal equations.
ComputeLeast Squares Solver using SVD
Solve the least squares problem using singular value decomposition.
Compute SolutionEigenvalues and Eigenvectors
Compute eigenvalues and eigenvectors.
ComputeSingular Value Decomposition (SVD)
Compute the full singular value decomposition.
Compute SVDMore 2D Visualizations
Determinant Visualizer
See the determinant as oriented area (2D parallelogram) or oriented volume (3D parallelepiped) formed by the column vectors of a matrix.
VisualizeMatrix-Vector Multiplication
Compute Ax interactively, display eigenvectors and eigenvalues for 2×2 matrices.
Visualize