Università Roma Tre

Part I: Linear algebra
The real n-dimensional vector space. Linear dependence and independence. Subspaces and spaces generated by vectors. Bases. Algebra of matrices. Determinant. Inverse matrix. Rank of a matrix. Systems of linear equations. Cramer's Theorem and Rouché-Capelli's Theorem. Homogeneous systems and linear dependence. Norm and Euclidean distance in . Scalar product. Sequences in . Topology and metrics in . Eigenvalues and eigenvectors. Diagonalization of matrices. Properties of eigenvalues.
Part II: Calculus of functions of several variables:
Continuous functions and Weierstrass theorem. Concave or convex functions. Partial derivatives and gradient. Differential. Derivations along a curve and directional derivative. Second order derivatives. Schwarz's theorem. Hessian matrix. Implicit function thoerem. Comparative statics. Inverse function theorem. Quadratic forms and matrices. Sign of a quadratic form.
Part III: Optimization:
Local and global maximum or minimum. First order conditions and second order conditions for free optimization. Equality constraints. Inequality constraints. Substitution method. Lagrange multiplier theorem. Second order conditions with constraints. Optimization for convex functions. Economic applications.
Part IV: Ordinary differential equations and systems
Definitions and examples. Exact differential. Equations in separable variables. Exact equations. Homogeneous equations. Malthusian growth model. Logistic growth model. Second order linear differential equations. General theorem of existence and uniqueness of the solution. Field of directions. Economic applications. Two-dimensional differential equation systems. Systems of linear differential equations: resolving method using eigenvalues, stationary states and their stability. Economic applications.

• Mastroeni L. e Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
or
• Simon & Blume: “Matematica per le scienze economiche” ed. Egea.