Università degli Studi di Roma "Tor Vergata"

Linear Algebra. Groups, fields, vector spaces. Linear independence and basis. Dimension of vector spaces. Linear transformations. Kernels. Scalar products. Cauchy-Schwartz inequality. Eigenvalues, eigenvectors and the characteristic polynomial of a square matrix. Basic properties of eigenspaces. Symmetric, skew-symmetric and orthogonal matrices. Positive definite matrices. Diagonalization

Probability. Elements of a probability space. Algebras of events and information about random experiments. Introduction to combinatorial calculus. Finite probability spaces, probability measures, introduction to Kolmogorov theory. Conditional probability, total probability formula, Bayes formula. Independent events. Random variables and their properties. Probability distribution, distribution function and densities function of a random variable. Expectation and variance of a random variable and their properties. Expectation and variance for the main kinds of random variables. Covariance and scale-invariance of the correlation coefficient. Random vectors and their properties. Probability distribution, distribution functions and densities functions of a random vector. Independent random variables, covariance and correlation. Conditional expectation of a random variable and its properties. Conditional expectation as best estimator. Geometric approach to the conditional expectation. Sequences of random variables. Convergence in probability and in law. The (weak) law of large numbers. The characteristic function. Central limit theorem. Multivariate Gaussian distribution.

Calculus. Series. Power series. The complex numbers. Complex power series and complex exponential. The Euler formula. Differentiability for functions of several variables: examples and counterexamples. The gradient. The Jacobian matrix. The chain rule for differentials. Mixed partial derivative. The Schwartz (Young) theorem. Integration in n-dimension. The Fubini theorem. The change of variable formula. Integration using polar coordinates. Differentiation under the integral sign. Introduction to differential equations. The Cauchy problem. Metric spaces. Normed spaces. Inner product spaces. The Sup norm on continuous functions. Pointwise convergence versus uniform convergence. The L^2 scalar product on R^2, on C[0,1] and for random variables. Density of polynomials in the space of continuous functions: the Bernstein proof of Weierstrass theorem by means of the weak law of large numbers.

Optimization. Taylor polynomial in n-dimensions. The Hessian matrix. Unconstrained optimization: necessary and sufficient conditions for maxima and minima. Constrained optimization. Lagrangian function and Lagrange multiplier. Introduction to Kuhn-Tucker