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Linear Algebra MicroSims

A collection of 126 interactive educational MicroSimulations for learning Applied Linear Algebra for AI and Machine Learning.

Vectors and Basic Operations

  • 2D/3D Vector Visualizer

    2D/3D Vector Visualizer Interactive visualization of vectors in 2D and 3D coordinate systems with adjustable components, projection lines, and component labels.

  • Vector Operations Playground

    Vector Operations Playground Interactive visualization of vector addition, subtraction, and scalar multiplication with draggable vectors and geometric constructions.

  • Row and Column Vectors

    Row and Column Vectors Interactive visualization comparing row vectors (horizontal, 1×n) and column vectors (vertical, m×1) to help students understand how orientation affects matrix operations.

  • Norm Comparison Visualizer

    Norm Comparison Visualizer Interactive visualization comparing L1 (Manhattan), L2 (Euclidean), and L-infinity (Maximum) norms through their unit shapes and distance measurements.

  • Dot and Cross Product

    Dot and Cross Product Interactive visualization comparing dot product (projection and angle) with cross product (perpendicular vector and parallelogram area).

  • Linear Combination Explorer

    Linear Combination Explorer Interactive visualization of linear combinations with adjustable coefficients, target challenges, and span visualization.

Vector Spaces

  • Vector Space Axiom Explorer

    Vector Space Axiom Explorer Interactive infographic for exploring the ten vector space axioms with hover definitions and concrete examples.

  • Vector Space Gallery

    Vector Space Gallery Interactive gallery showcasing six diverse vector space examples with visual representations, zero vectors, and operation examples.

  • Subspace Tester

    Subspace Tester Interactive visualization to test whether sets are subspaces by checking closure under linear combinations.

  • Inner Product Visualizer

    Inner Product Visualizer Interactive visualization showing how different inner products define different notions of length and angle.

Matrices

  • Matrix Basic Operations

    Matrix Basic Operations Interactive visualization demonstrating element-wise matrix addition and scalar multiplication with step-by-step calculation highlighting.

  • Matrix Multiplication

    Matrix Multiplication Step-by-step visualization of matrix multiplication showing row-by-column dot product calculations with animation and highlighting.

  • Matrix Inverse

    Matrix Inverse Interactive exploration of 2×2 matrix inversion with real-time computation, verification that AA⁻¹ = I, and visualization of singular matrices.

  • Special Matrices

    Special Matrices Visual gallery of special matrix types including identity, diagonal, upper triangular, and lower triangular matrices with interactive size control.

  • Symmetric Matrix

    Symmetric Matrix Interactive visualization demonstrating symmetric matrices where A[i,j] = A[j,i], with adjustable size from 2×2 to 10×10.

  • Sparse and Dense Matrices

    Sparse and Dense Matrices Side-by-side comparison of sparse and dense matrices showing structural differences, storage efficiency, and common sparsity patterns.

  • Block Matrix

    Block Matrix Interactive visualization of matrix partitioning into blocks with draggable partition lines showing how large matrices can be decomposed into submatrices.

Systems of Linear Equations

  • System Geometry

    System Geometry Interactive visualization showing how systems of linear equations correspond to geometric intersections of lines (2D) or planes (3D).

  • Gaussian Elimination

    Gaussian Elimination Step-by-step animated guide through the Gaussian elimination algorithm with explanations.

  • Row Operations

    Row Operations Interactive practice tool for applying elementary row operations on augmented matrices.

  • REF vs RREF

    REF vs RREF Side-by-side comparison of Row Echelon Form and Reduced Row Echelon Form with highlighted differences.

  • Solution Sets

    Solution Sets Explore how different systems produce unique solutions, infinite solutions (lines/planes), or no solution.

  • Homogeneous Systems

    Homogeneous Systems Explore homogeneous systems Ax = 0 and visualize their null spaces as subspaces through the origin.

  • Numerical Stability

    Numerical Stability Explore how ill-conditioned systems amplify small input errors into large solution changes.

Linear Transformations

  • Linear Transform Basics

    Linear Transform Basics Interactive visualization showing how linear transformations preserve grid structure and are determined by where basis vectors map.

  • Transform Gallery

    Transform Gallery Compare and contrast rotation, scaling, shear, and reflection transformations with interactive controls and live matrix display.

  • Transform Composition

    Transform Composition Demonstrate that the order of transformations matters by comparing T then S versus S then T side by side.

  • 2D Rotation

    2D Rotation Interactive visualization of 2D rotation matrices showing the relationship between rotation angle and cos/sin matrix entries.

  • Orthogonal Transform

    Orthogonal Transform Interactive visualization demonstrating how orthogonal matrices (rotations and reflections) preserve lengths and angles when transforming shapes.

  • Orthogonal Projection

    Orthogonal Projection Visualize how vectors project onto lines with perpendicular error vectors and live formula display.

  • Kernel and Range

    Kernel and Range Visualize the kernel (null space) and range (column space) of linear transformations, demonstrating the rank-nullity theorem.

  • Change of Basis

    Change of Basis Visualize how the same vector has different coordinate representations in different bases, with transition matrix display.

  • Basis Coordinate Visualizer

    Basis Coordinate Visualizer Side-by-side comparison of standard and custom basis coordinate systems showing how the same vector has different coordinate representations.

Determinants

  • 2×2 Determinant Calculator

    2×2 Determinant Calculator Interactive calculator for computing 2×2 determinants with step-by-step visualization and geometric interpretation.

  • Signed Area

    Signed Area Visualize the signed area of a parallelogram formed by two vectors, showing how orientation affects the sign of the determinant.

  • Sarrus Rule

    Sarrus Rule Step-by-step animation showing how to compute 3×3 determinants using the Rule of Sarrus.

  • Cofactor Expansion

    Cofactor Expansion Step-by-step animation showing cofactor expansion for computing determinants of any size matrix.

  • Determinant Properties

    Determinant Properties Interactive exploration of how row operations affect determinant values.

  • Singular Matrix

    Singular Matrix Visualize the geometric difference between singular and non-singular matrices through transformation animation.

  • Cramer's Rule

    Cramer's Rule Step-by-step visualization of solving systems of equations using Cramer's Rule with determinants.

  • Volume Scaling 3D

    Volume Scaling 3D Visualize how 3×3 matrix transformations scale 3D volumes, connecting the determinant to geometric volume change.

Eigenvalues and Eigenvectors

  • Eigenvector Transformation

    Eigenvector Transformation Interactive visualization demonstrating how eigenvectors maintain their direction under linear transformation while other vectors change direction.

  • Characteristic Polynomial

    Characteristic Polynomial Interactive computation of characteristic polynomials and eigenvalues for 2×2 and 3×3 matrices with graphical visualization.

  • Eigenspace Visualization

    Eigenspace Visualization 3D visualization of eigenspaces showing how geometric multiplicity determines whether eigenspaces are lines or planes through the origin.

  • Multiplicity Comparison

    Multiplicity Comparison Compare algebraic and geometric multiplicity across different matrix types to understand diagonalizability conditions.

  • Diagonalization Workflow

    Diagonalization Workflow Interactive flowchart guiding through the step-by-step process of diagonalizing a matrix with decision points.

  • Matrix Power Calculator

    Matrix Power Calculator Demonstrates how diagonalization simplifies computing matrix powers using the eigenvalue decomposition A^k = PD^kP⁻¹.

  • Complex Eigenvalue

    Complex Eigenvalue Visualize how complex eigenvalues λ = a + bi correspond to rotation-scaling transformations in 2D.

  • Spectral Theorem

    Spectral Theorem Interactive demonstration of the spectral theorem showing how symmetric matrices decompose into orthogonal eigenvectors and real eigenvalues.

  • Power Iteration

    Power Iteration Visualization of the power iteration method for finding the dominant eigenvalue and eigenvector through repeated matrix-vector multiplication.

  • Eigenvalue Applications

    Eigenvalue Applications Interactive hub-and-spoke infographic showing how eigenanalysis concepts connect to real-world applications in machine learning, AI, and science.

Orthogonality and Least Squares

  • Gram-Schmidt Process

    Gram-Schmidt Process Step-by-step 3D visualization of Gram-Schmidt orthonormalization showing projections and construction of orthonormal vectors.

  • Gram-Schmidt (Detailed)

    Gram-Schmidt (Detailed) Detailed step-by-step 3D visualization of Gram-Schmidt orthonormalization showing projection computation, subtraction, and normalization phases.

  • Orthonormal Basis Finder

    Orthonormal Basis Finder Interactive visualization demonstrating how orthonormal bases simplify coordinate finding through inner products.

  • Projection onto Subspace

    Projection onto Subspace 3D visualization of vector projection onto subspaces showing the projection as the closest point and the orthogonal error vector.

  • Least Squares Visualizer

    Least Squares Visualizer Interactive visualization of least squares as projection showing the geometric relationship between b, Ax-hat, and the error vector with dual regression and geometric views.

  • Four Subspaces

    Four Subspaces Visualize the four fundamental subspaces of a matrix and their orthogonal relationships, demonstrating the Fundamental Theorem of Linear Algebra.

  • Pseudoinverse Solver

    Pseudoinverse Solver Interactive exploration of the Moore-Penrose pseudoinverse for solving least squares problems, including overdetermined, underdetermined, and rank-deficient systems with SVD visualization.

Matrix Decompositions

  • LU Decomposition

    LU Decomposition Step-by-step visualization of LU decomposition showing how Gaussian elimination produces L and U matrices.

  • Matrix Rank Visualizer

    Matrix Rank Visualizer Interactive 3D visualization showing how matrix rank relates to the column space, with column vectors displayed geometrically and row echelon form computation.

  • Positive Definiteness

    Positive Definiteness Interactive 3D visualization of quadratic forms showing how eigenvalue signs determine positive definiteness.

  • SVD Geometry

    SVD Geometry Visualize SVD as a sequence of rotation-scaling-rotation transformations on the unit circle.

  • SVD Forms Comparison

    SVD Forms Comparison Visual comparison of Full, Compact, and Truncated SVD showing matrix dimensions and storage requirements.

  • SVD Image Compression

    SVD Image Compression Interactive demonstration of image compression using truncated SVD.

  • Condition Number

    Condition Number Visualize how condition number affects the sensitivity of linear system solutions to perturbations.

  • Decomposition Guide

    Decomposition Guide Interactive decision tree for choosing the right matrix decomposition based on your problem.

Data Science Applications

  • Data Matrix Structure

    Data Matrix Structure Interactive visualization showing the structure of data matrices with rows as samples and columns as features, including heat map coloring.

  • Covariance and Correlation

    Covariance and Correlation Interactive exploration of how covariance and correlation capture relationships between features through scatter plots and heatmaps.

  • PCA Explorer

    PCA Explorer Interactive visualization demonstrating Principal Component Analysis step by step, from raw data through centering, eigenvector computation, and projection.

  • Scree Plot

    Scree Plot Interactive visualization for learning to use scree plots and cumulative variance to select the optimal number of principal components.

  • Linear Regression

    Linear Regression Interactive visualization of linear regression showing how the best-fit line minimizes squared errors with draggable data points and loss surface heatmap.

  • Regularization Geometry

    Regularization Geometry Interactive visualization showing how L1 and L2 regularization constrain model weights geometrically, demonstrating why L1 produces sparse solutions.

Neural Networks

  • Neural Network Layer

    Neural Network Layer Interactive visualization of a neural network layer showing how matrix-vector multiplication implements the forward pass with various activation functions.

  • Neural Network Architecture

    Neural Network Architecture Interactive visualization of neural network architecture showing layers, neurons, weight matrix dimensions, and parameter counts.

  • Activation Functions

    Activation Functions Interactive comparison of neural network activation functions including ReLU, Sigmoid, Tanh, Leaky ReLU, and Softplus with derivative visualization.

  • Perceptron Decision Boundary

    Perceptron Decision Boundary Interactive visualization showing how perceptron weights and bias define a linear decision boundary for binary classification.

  • Forward Propagation

    Forward Propagation Step-by-step visualization of forward propagation through a neural network showing matrix operations at each layer.

  • Backpropagation

    Backpropagation Step-by-step visualization of backpropagation showing how gradients flow backward through a neural network via the chain rule.

  • Normalization Comparison

    Normalization Comparison Visual comparison of batch normalization and layer normalization showing which tensor dimensions each technique normalizes.

  • Tensor Operations

    Tensor Operations Interactive visualization of common tensor operations including reshape, transpose, flatten, squeeze, and unsqueeze.

Optimization

  • Gradient Descent

    Gradient Descent Visualize how gradient descent optimization navigates loss surfaces and how learning rate affects convergence behavior.

  • Learning Rate Effect

    Learning Rate Effect Interactive side-by-side comparison showing how different learning rates affect gradient descent optimization, demonstrating convergence, oscillation, and divergence.

  • ML Pipeline

    ML Pipeline Interactive flowchart showing the complete ML pipeline from raw data to trained model with code examples and detailed explanations.

  • Convex Function Visualizer

    Convex Function Visualizer Interactive visualization of convex functions and their properties for optimization.

  • Newton vs Gradient Descent

    Newton vs Gradient Descent Side-by-side comparison of Newton's method and gradient descent showing convergence characteristics.

  • Hessian Curvature Visualizer

    Hessian Curvature Visualizer Interactive visualization of how the Hessian matrix captures surface curvature for optimization.

  • SGD Trajectory Visualizer

    SGD Trajectory Visualizer Visualize stochastic gradient descent trajectories and noise characteristics.

  • Momentum Dynamics Visualizer

    Momentum Dynamics Visualizer Interactive visualization of momentum in optimization showing how it helps overcome local minima.

  • Optimizer Comparison Arena

    Optimizer Comparison Arena Compare different optimization algorithms side-by-side on the same loss surface.

  • Lagrange Multiplier Geometry

    Lagrange Multiplier Geometry Interactive visualization of Lagrange multipliers for constrained optimization.

  • KKT Conditions Visualizer

    KKT Conditions Visualizer Visualize the Karush-Kuhn-Tucker conditions for constrained optimization problems.

Transformers and NLP

  • Attention Mechanism

    Attention Mechanism Interactive visualization of the attention mechanism used in transformer models.

  • Multi-Head Attention

    Multi-Head Attention Visualize how multi-head attention allows the model to attend to different positions simultaneously.

  • Transformer Block

    Transformer Block Interactive visualization of a complete transformer block showing self-attention and feed-forward layers.

  • Embedding Space Visualizer

    Embedding Space Visualizer Explore word and document embeddings in high-dimensional space projected to 2D/3D.

  • Cosine and Euclidean Similarity

    Cosine and Euclidean Similarity Compare cosine similarity and Euclidean distance for measuring vector relationships.

  • LoRA Visualizer

    LoRA Visualizer Visualize Low-Rank Adaptation (LoRA) for efficient fine-tuning of large language models.

  • Latent Space Interpolation

    Latent Space Interpolation Explore smooth transitions in latent space by interpolating between embeddings.

Computer Vision

  • Image Matrix Visualizer

    Image Matrix Visualizer Interactive visualization showing how pixel values in a matrix correspond to grayscale image appearance with hover highlighting and edit mode.

  • RGB Channel Decomposition

    RGB Channel Decomposition Interactive visualization showing how RGB color channels combine to form color images with channel isolation and intensity controls.

  • Convolution Operation

    Convolution Operation Interactive visualization of the convolution operation showing how kernels slide across images with stride and padding controls.

  • Convolution Visualizer

    Convolution Visualizer Step-by-step visualization of image convolution showing how kernels slide across images to compute filtered outputs with multiple kernel types.

  • Filter Effects Gallery

    Filter Effects Gallery Side-by-side comparison of image filters including blur, sharpen, edge detection, and emboss effects with kernel visualization.

  • Edge Detection Visualizer

    Edge Detection Visualizer Interactive visualization of Sobel, Prewitt, and Scharr edge detection operators showing gradient components, magnitude, and thresholded edges.

  • Corner Detection Visualizer

    Corner Detection Visualizer Interactive Harris corner detection visualization showing structure tensor eigenvalue analysis and response heatmaps.

  • Fourier Transform Visualizer

    Fourier Transform Visualizer Interactive 2D Discrete Fourier Transform visualization showing spatial-frequency relationship, magnitude/phase spectra, and frequency filtering.

  • Homography Demo

    Homography Demo Interactive demonstration of perspective transformations using homography matrices with draggable corner points and real-time matrix computation.

  • SVD Compression Visualizer

    SVD Compression Visualizer Interactive demonstration of image compression using truncated Singular Value Decomposition with quality metrics.

3D Vision and Geometry

  • Camera Model Visualizer

    Camera Model Visualizer Interactive demonstration of the pinhole camera model showing how intrinsic parameters affect 3D-to-2D projection.

  • Camera Calibration

    Camera Calibration Interactive demonstration of camera calibration showing lens distortion effects, checkerboard corner detection, and distortion correction.

  • Epipolar Geometry

    Epipolar Geometry Interactive demonstration of epipolar constraints in stereo vision showing epipolar lines, planes, and depth from disparity.

  • Triangulation Visualizer

    Triangulation Visualizer Interactive demonstration of 3D point recovery from stereo correspondences showing triangulation accuracy and noise effects.

  • Point Cloud Visualizer

    Point Cloud Visualizer Interactive exploration of point cloud data with different datasets, color modes, downsampling, and surface normal visualization.

Autonomous Systems

  • Coordinate System 3D

    Coordinate System 3D Interactive demonstration of different 3D coordinate system conventions and handedness including OpenGL, DirectX, ROS, and camera frames.

  • Euler Angles Visualizer

    Euler Angles Visualizer Interactive demonstration of how Euler angles (yaw, pitch, roll) compose to form 3D rotations with multiple conventions and gimbal lock warning.

  • Gimbal Lock Demo

    Gimbal Lock Demo Interactive demonstration of gimbal lock using a physical gimbal mechanism with three nested rings showing loss of degree of freedom.

  • Quaternion Rotation

    Quaternion Rotation Interactive demonstration of quaternion rotation representation with axis-angle conversion, rotation application, and composition.

  • Rigid Body Transform

    Rigid Body Transform Interactive visualization of rigid body transform composition in a robot arm kinematic chain showing forward kinematics.

  • Kalman Filter

    Kalman Filter Interactive visualization of the Kalman filter showing the predict-update cycle, uncertainty propagation, and noise effects on state estimation.

  • Sensor Fusion

    Sensor Fusion Interactive demonstration of GPS and IMU sensor fusion using Kalman filtering, showing how combining complementary sensors improves accuracy.

  • LIDAR Point Cloud

    LIDAR Point Cloud Interactive 3D visualization of LIDAR point cloud data demonstrating ground segmentation, object clustering, and coloring modes.

  • SLAM Visualizer

    SLAM Visualizer Interactive visualization of Simultaneous Localization and Mapping showing robot trajectory, landmark mapping, and loop closure optimization.

  • Object Tracking

    Object Tracking Interactive multi-object tracking demonstration showing the predict-associate-update cycle with bounding boxes, track IDs, and data association.

  • Path Planning

    Path Planning Interactive comparison of path planning algorithms (A*, Dijkstra, RRT) showing exploration patterns, path quality, and performance metrics.

  • Trajectory Optimization

    Trajectory Optimization Interactive trajectory optimization demonstration showing smoothness constraints, velocity limits, obstacle avoidance, and cost convergence.

Learning Tools

  • Learning Graph Viewer

    Learning Graph Viewer Interactive visualization of the course concept dependency graph with 300 concepts and their relationships.