Applied Linear Algebra Interactives

Nine small web apps that let you see and test the ideas of linear algebra, from linear systems to a tiny neural network. Move a slider, drag a vector or type a matrix, and watch the picture and the numbers change together. Each one works with a keyboard, a mouse or a touch screen, describes its picture in text, and has built-in challenges with instant feedback.

Linear systems

How Many Solutions? Lines and planes Live

Change the coefficients of two or three equations and watch lines in the plane, or planes in space, meet in one point, along a line, or not at all; the augmented matrix and its row-reduced form update live.

Row Reduction Workbench Live

Type a matrix (up to 4x6), apply row operations with exact fraction arithmetic, undo, and get told when you reach REF and RREF; a step that undoes progress gets a warning, and a missed elimination a tip with the right multiplier.

Vectors and transformations

Span Builder Live

Choose two or three vectors in the plane or in space, drive the coefficients with sliders to hit a target vector, and see the span: a line, a plane or everything.

Matrix Transformer Live

Set a 2x2 or 3x3 matrix and watch it act on the unit square, circle, cube or sphere: the columns, the signed area or volume (determinant), the ellipse half-axes (singular values), and the motion from I to A.

Eigenvalues

Eigenvector Hunt Live

Rotate a unit vector x and watch Ax; find the directions where Ax is parallel to x, read off lambda, then iterate x -> Ax (power method) to see the dominant eigenvector emerge.

Geometry: norms and projections

Norms and Inner Products: Unit Balls Live

See the unit balls of the 1-, 2- and max-norms and of the A-norm side by side, with the length of your own vector x in each; find out when a symmetric matrix A gives a norm (positive definiteness).

Projection and Least Squares Lab Live

Project a vector onto a line and watch the error stay perpendicular; fit a line to data with the normal equations; then fit any basis of functions, including your own, and compare the errors.

Optimization and learning

Gradient Descent Playground Live

Minimize a quadratic bowl f(x) = 1/2 x^T A x - b^T x from a starting point; change the learning rate and watch convergence, zig-zag, or divergence; stopping criteria shown live.

Tiny Neural Network: XOR Live

Train a small XOR network (2 inputs, a sigmoid hidden layer of 2, a linear output, mean squared error) with backpropagation; step through one forward and backward pass with every number shown, watch the loss plateau and the decision boundary form.