Linear systems, examples. Naive matrix representation, Gauss elimination. Linear functions, matrices. Vector spaces. Linear independence. Rank of a matrix, computation via Gauss elimination. Basis. Scalar product in R^n. Linear transformations between vector spaces and their matrix representation. Matrix-vector product. Image of basis vectors. Kernel and image of a linear transformation. Injectivity. Counting dimensions of kernel and image. Geometric interpretation of a linear system of equations. Volume transformation in R^n via linear transformations. Determinants. Examples in R^2 and R^3. Vector product. Multilinearity of determinants. Computation of determinants via Gauss elimination. Matrix singularity. Determinantal formulae: expansion via permutations, Laplace expansion. Equivalence classes of matrices. Diagonal representation. Intrinsic notion of eigenvector and eigenvalue. Explicit representation: characteristic equation, roots, algebraic multiplicity. Eigenspaces, geometric multiplicity. Diagonalization. Cayley-Hamilton theorem. Matrix calculus. Symmetric matrices and orthogonal transformations.
Multivariable calculus. Basic notions, continuity, partial derivatives. Differential as a linear transformation. Gradient, level curves. Calculus of derivatives. Implicit function theorem. Critical points, maxima and minima, saddle points. Taylor polynomial in many variables. Quadratic forms. Lagrange multipliers. Parametric curves and surfaces.
Integrals of functions of many variables. Flow, circulation, Gauss Green theorem. Basic calculus techniques. Change of variables, jacobian.
Ordinary differential equations. Classification and basic properties. Linear equations with constant and variable coefficients. Second order equations with constant coefficoents. Order reduction. Variation of constants and variation of parameters. Non-linear equations of first order. Phase space. Vector fields. Separable equations, exact equations. Integrating factor.