Program | Texts |

Diary of lessons of the part of this course concerning linear algebra and 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)
**

** Thursday 09/27/2012**

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

- Example: {v in
**R**^{n}such that ||v||= 1} - 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
- Other scalar products in
**R**^{n} - The problem of conditioning
- Matrix norms
- The fundamental inequality

- Calculation of matrix norms in several ways
- Analytic geometry
- Curves and surfaces: cartesian and parametric representation

- Ruled surfaces
- Cylinders, cones, surfaces of revolution
- Quadrics: cylinders, cones and ellipsoides

- Quadrics: Hyperboloids and paraboloids
- Symmetry planes, center, tangent plane
- Doubly ruled quadrics

- Changes of coordinates in plane and in space
- Criteria for recognizing quadrics
- Tangent planes
- Kinds of points
- Construction of quadrics

- Sample of exam:

- Calculation of condition number

- Projection onto subspace

- Study of quadrics

- Changes of coordinates

**Theory notes **

Outline of theory. First part: Orthonormal bases, norms and condition number: | 7 pages
| Sep 27, 2013 |

Outline of theory. Second part: Changes of coordinates, curves, surfaces, quadrics | 8 pages
| Nov 5, 2012 |

**Exercises (with many answers) **

**Warning**: These
notes are under further development:

A new version will be ready in a few weeks

Exercises on Orthonormal bases, norms and condition number | 2 pages
| Dec 23, 2012 |

Answers to Orthonormal bases, norms and condition number | 3 pages (for now)
| Dec 23, 2012 |

Exercises on Changes of coordinates, curves, surfaces, quadrics | 5 pages
| Dec 23, 2012 |

Answers to Changes of coordinates, curves, surfaces, quadric | 6 pages (for now)
| Dec 23, 2012 |

Comments and notifications of errors shall be welcome.

At the bottom of each page you'll find the date of release.