The Invertible Matrix Theorem is the master key to linear algebra. This visual lesson organizes 50 equivalent conditions connecting pivots, row reduction, rank, null spaces, bases, linear systems, one-to-one and onto transformations, determinants, eigenvalues, transposes, singular values, Gram matrices, and conditioning—with proofs, examples, and a practical mastery map.
The Jacobian is the hidden scale factor behind every coordinate change in Calculus III. In this Woody Calculus visual lesson, learn how Jacobians explain area and volume scaling, change of variables, polar coordinates, cylindrical coordinates, spherical coordinates, double integrals, triple integrals, and why the mysterious extra factors r and ρ²sinφ appear.
The determinant is the number that measures how a matrix changes space. In this Woody Calculus visual lesson, learn how determinants explain area and volume scaling, orientation, invertibility, matrix collapse, eigenvalues, Wronskians, and why det(A) connects Linear Algebra to Differential Equations.