Linear Algebra simulators
Vectors, matrices and the transformations they encode.
Linear Transformations
Drag the matrix entries and watch the plane deform with them, carrying the grid and the two basis vectors along for the ride.
Matrix Multiplication
Multiplying two matrices chains their maps: AB means apply B, then A. Run both orders side by side and watch the letter F end up in two different places.
The Determinant
One number for what a matrix does to area. Squash the parallelogram to nothing and the map has thrown a dimension away; push past that and the plane comes back inside out.
Systems of Equations
Two equations in two unknowns, seen two ways at once: as lines that cross, and as columns that add up to b. Elimination turns the lines about the answer without ever moving it.
Column Space & Null Space
A matrix squeezes its input space onto its column space and crushes its null space to zero. Drag a point along the null line and its image never moves; pull a column into the plane of the others and watch rank fall as nullity rises.
Change of Basis
The same arrow, watched through two different grids and written down as two different pairs of numbers. The dictionary between them is a matrix, the pairs it can reach are the span of the basis, and a transformation translated is the sandwich.
Eigenvectors & Eigenvalues
A matrix knocks almost every vector off its own line. Aim v until Av lands back on it: that is an eigenvector, and the stretch is λ.
Dot Product
One length times the shadow the other casts on it, with the component formula falling straight out of that one picture.
Projections & Gram–Schmidt
The closest point of a plane to a vector is where the error meets it at a right angle. The same picture fits a line to data, and done one direction at a time it straightens any basis into an orthonormal frame, A = QR.
Cross Product
Two vectors span a patch of area. Its size is the length of the answer, its tilt is the direction, and each component is a shadow of that patch.
Singular Value Decomposition
Watch any matrix turn a circle into an ellipse as rotate, stretch, rotate, the geometry of A = U Σ Vᵀ. Then keep only the biggest singular values and compress a picture.