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.
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.
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, and a transformation translated is the sandwich.
Eigenvectors & Eigenvalues
Spin a vector around the circle and watch where the matrix sends it. When the input and output line up, you have found an eigenvector.
The Dot Product
One length times the shadow the other casts on it, with the component formula falling straight out of that one picture.
The 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.
Span
Drag two vectors and see everywhere their scaled sums can reach, the whole plane, or a single line when they line up.
Singular Value Decomposition
Watch any matrix turn a circle into an ellipse as rotate, stretch, rotate, the geometry of A = U Σ Vᵀ.