Explore matrix calculus in this comprehensive lecture from MIT's Linear Algebra course. Delve into scalar calculus, linearization, gradients, and geometric interpretations. Learn about matrix/vector product rules and sophisticated gradient calculation methods. Examine specific examples, including f(x) = (Ax-b)'(Ax-b), and understand gradient notation and the trace concept. Investigate linear functions of matrices, gradients of matrix-to-scalar functions, and vector-to-vector Jacobians. Discover practical applications of gradients and explore Jacobian matrices for vector-to-vector and matrix-to-matrix functions. Gain insights into the relationship between Jacobians and volumes, and learn why writing out matrix elements isn't always necessary.