The module of "Mathematics" of this course is divided into two parts:

The first part held by prof. F. Odetti deals with Linear Algebra and Analytic Geometry.
The second part held by prof. E. Mainini deals with Analysis and Calculus.

In this web page we deal with the first part of Linear Algebra and Analytic Geometry

Prerequisites: Linear algebra: matrices, linear systems, gaussian elimination.
Vector spaces: Subspaces, linearly independent vectors, generators (spanning sets), bases, dimension.
Diagonal form of a matrix: eigenvalues, eigenvectors, eigenspaces.
Scalar products. Orthogonal vectors. Orthonormal bases. Orthogonal matrices.
Spectral theorem (symmetric matrices are diagonalizable). Quadratic forms.
Analytic geometry of plane: conic sections.
Analytic geometry of space: lines, planes, distances, circles, spheres.

An outline of the program is the following:

• Complements of linear algebra: non-finitely generated vector spaces
• Oorthonormal bases, projections, condition number of a matrix.
• Complements of analytic geometry: curves and surfaces.
• Changes of coordinates.
• Quadrics.

References:

• for linear algebra and geometry

    Odetti-Raimondo - Elementi di algebra lineare e geometria analitica (in italian)
    Gilbert Strang - Linear algebra (in english)

• for calculus

    R. A. Adams - Calcolo Differenziale 1 (numerical integration, series) both in italian and english
    R. A. Adams - Calcolo Differenziale 2 (triple ntegrals and surface integrals) both in italian and english
    C.D. Pagan - S. Salsa - Analisi Matematica 2 (in italian)
  

Lessons of geometric and algebraic part (prof. F.Odetti)

Tuesday 09/23/2014

• Recalls about vector spaces
• Bases of vector spaces
• Hamel bases. Example: monomials in R[x]
• Outline of Hilbert bases
• Scalar products
• Euclidean product in Rⁿ
• Norms
• Norms in Rⁿ: 1-norm, p-norm, inf-norm
• Scalar product in C^o[a,b] using integrals
• Norms in C^o[a,b]
• Example: {v in Rⁿ such that ||v||= 1}
• Cauchy-Schwarz inequality

• Algorithm of Gram-Schmidt
• Example: orthonormal basis of polynomials in C^o[-1,1]
• Projection of a vector into a subspace and its properties.
• Example: {sin(x), cos(x), sin(2x),...} is a o.n. basis and related to Fourier series
• Projections
• Orthogonal matrices
• Spectral theorem

• Definite positive matrices
• Sylvester's law of inertia
• The problem of conditioning in Linear Algebra
• Matrix norms
• The fundamental inequality of conditioning
• Well-conditioned and ill-conditioned matrices
• Calculation of matrix 2-norm
• Singular values

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