m Geometria II 2011-'12 - Ing. Meccanica

Prof. Giuseppe Pareschi


Department of Mathematics

Viale della Ricerca Scientifica 1, 00133, Roma, IT

Stanza: 0212

Telefono: 06 72594621

pareschi@mat.uniroma2.it





LINEAR ALGEBRA AND GEOMETRY, YEAR 2011-'12

Instructor: Prof. Giuseppe Pareschi

Teaching Assistant: Dr. Pietro Sabatino (PhD)

Timetable course :
MON 2.00 - 3.45, room 7,
TUE 11.30 - 1.15 pm, room 7,
FRI 2.00 - 3.45, room 7,

Timetable tutoring (Dr. Pietro Sabatino):
WED 9.30 - 11.15, room 8

ANNOUNCEMENTS (IMPORTANT!):
NEW! In order to avoid overlapping with other exams of the first year, the written exam of the VI session is moved up. The new date (see below) is Tuesday feb 26, at 11.00. Room B11

Office hours:
After the lectures and by appointment (send me an e-mail)

Exams:
The exam consiste of a written test and of an oral examination. We will have:
  • one session at the end of june, in the week after the end of the course;
  • one session in july;
  • two sessions in september;
  • two sessions is february 2013.


Timetable exams :
  • 1st Session - Summer: FRI July 6, at 11.15. Room B8 (written). MON July 9, at 11.30, room B8 (oral)
  • 2nd Session - Summer: FRI July 20, at 11.15. Room B8 (written). Oral: MON july 23, at 1.00 p.m., (and NOT, as previously written, at 1.30 pm) room B8 .
  • 3rd Session - Autumn: WED Sept 12, at 11.15, room B8 (written). NOTE: The date may change. Check the date at the end of august. Oral: to be decided.
  • 4rd Session - Autumn: MON Sept 24, at 11.15, room B8 (written). NOTE: The date may change. Check the date at the end of august. Oral: to be decided.
  • 5th Session - Winter: WED Feb 13, at 11.00, room B9 (written test). Warning: check the date a few days before the exam. Oral test: to be decided.
  • 6th Session - Winter: NEW! WED Feb 26, at 11.00, room B11 (written test). Warning: check the date a few days before the exam. Oral test: to be decided.


Exam results : Textbook: T. Apostol, Calculus, Vol. I and II.
Vol. I: Chapters 12, 13, 14, 15, 16.
Vol. II: Chapters 3, 4, 5.

Syllabus: pdf

Text and solutions of last year written tests

Weekly description of the topics of the lectures, with assigned exercises
  • Week 1 (March 5 - 9):
    Topics: The space of real n-tuples. Vector addition. Scalar multiplication. Dot product. Norm. Orthogonality.
    Reference: Apostol, Vol. I, Sections from 12.1 to 12.8.
    Assigned exercises: Sections 12.4 and 12.8.

  • Week 2 (March 12 - 17):
    Topics: Unit vectors. Distance between two n-tuples. Orthogonal projection and orthogonal decomposition. Angle between to vectors in V(n). The linear span of a finite subset of vectors of V(n). Linear independence.
    Reference: Apostol, Vol. I, Sections from 12.9 to 12.13.
    Assigned exercises: Sections 12.11 (whole) and 12.14 (up to exercise 16).

  • Week 3 (March 19 - 24):
    Topics: Linear independence (cont.). Bases of V(n). Components of a vector with respect to a basis. Components of a vector with respect to orthogonal and/or orthonormal bases. Lines in V(n). Different parametric equations of the same line. Parallel lines. Intersection of lines. Distance point-line.
    Reference: Apostol, Vol. I, Sections 12.14, and from 13.1 to 13.4.
    Assigned exercises: Sections 12.14: from n. 17 to n. 20. Section 13.5 (whole).

  • Week 4 (March 26 - 30):
    Topics: Normal vector to a line in V(2) and cartesian equation. Distance point-line in V(2). Planes. Different parametric representations of the same plane. Plane through three non-collinear points. Determinant of a square matrix of order three. Cross product of two vectors in V(3). Area of a parallelogram in V(3). Mixed product (scalar triple product). Mixed product and linear independence. Volume of a parallelepiped in V(3). Cramer's rule for systems of three linear equations in three unknowns. Normal vector to a plane in V(3) and cartesian equation.
    Reference: Apostol, Vol. I, Sections from 13.6 to 13.15.
    Assigned exercises: Sections 13.8, 13.11, 13.14. Section 13.17: only from n.1 to n. 6.

  • Week 5 (only April 2 and 3):
    Topics: Distance point-plane in V(3). Conic sections: definition via eccetricity. Fundamental equation. Focal (or polar) equation. Conic sections with central symmetry: ellipses and hyperbolas. Equations and geometric properties. Parabolas.
    Reference: Apostol, Vol. I, Sections 13.15 and 13.16. Sections from 13.19 to 13.23.
    Assigned exercises: Sections 13.17, 13.21, 13.24. Section 13.25, except n. 17, 20, 24.

  • Week 6 (only April 13):
    Topics: Exercises on planes and conic sections.

  • Week 7 (only April 16 - 21):
    Topics: Vector-valued functions of real variable: basic operations, derivatives and integrals. Curves described by vector-valued functions. Tangent line. Example: reflection properties of conic sections. Velocity vector, speed, unit tangent vector, acceleration vector, unit normal vector. Osculating plane. Plane curves and the angle of inclination of the velocity vector. Arc-length and arc-length functions. Arc-length reparametrization. Curvature. Examples: characterization of straight lines and circles. Circular helices.
    Reference: Apostol, Vol. I, Sections from 14.1 to 14.3 and sections 14.5, 14.6, 14.8, Section from 14.10 to 14.12 (only the arc-length and the arc-length function defined as integrals, and the fact that the arc-length function is a primitive of the speed function). Section 4.14.
    Assigned exercises: Sections 14.4, 14.7, 14.13, 14.15.

  • Week 9 (only fri, May 4):
    Topics: Complex numbers. Definition. Addition and multiplication. Field axioms. Square root of a negative real number. Solutions of real quadratic equations. Solutions of polynomial equations. Fundamental Theorem of Algebra. Complex numbers in polar coordinates. n-th roots of 1.
    Reference: Apostol, Vol. I, Sections from 9.1 to 9.5
    Assigned exercises: Section 9.6

  • Week 10 (May 7 and May 11):
    Topics: Complex numbers: real part, imaginary part, conjugate, modulus. Complex and real solutions of real polynomial equations. The space of complex n-tuples: addition, scalar multiplication, dot product. Linear spaces: axioms and examples.
    Reference: Apostol, Vol. I, Section 12.16, from section 15.1 to section 15.4 and 15.6
    Assigned exercises: Section 9.10 n.5 and n.8, Section 12.17 except n.7 and n.8, Section 15.4

  • Week 11 (may 14 -18):
    Topics: Linear combinations, the linear span of a finite set of elements of a givel linear space, linear independence/dependence, finite/infinite dimensional linear spaces, dimension and bases of a finite-dimensional linear space, inner products in the real and in the complex case, Cauchy-Schwartz inequality, norm, distance, perpendicularity, orthogonal sets, orthogonal/orthonormal bases, components of an element with respect to an oprthonormal basis, Gram-Schmidt orthogonalization.
    Reference: Apostol, Vol. I, Sections from 15.7 to 15.9, Sections 15.11, 15.12 and Section 15.14
    Assigned exercises: Section 15.10, 15.13

  • Week 12 (may 21 -25):
    Topics: The orthogonal complement of a linear subspace. The Orthogonal Decomposition Theorem. Orthogonal projection of an element on a finite-dimensional subspace. Examples. Linear transformations. Linear transformations with prescribed values on a basis of the domain. Matrices: linear spaces of matrices, matrix multiplication.
    Reference: Apostol, Vol. I, Sections 15.14 and 15.15. Sections 16.1, 16.9, 16.13, 16,15
    Assigned exercises: Section 15.16, except n. 5. Section 16.16. More exercises.

  • Week 13 (May 27 to june 1):
    Topics: Matrix associated to a linear transformation from V(n) to V(m) in the canonical bases. Matrix representing a linear trasformation with respect to two bases (one of the source and of the target). Review of elementary properties of functions: injectivity, surjectivity, invertibility and inverse function, bijectivity. Null-space and range of a linear transformation. Null-space and injectivity. The "nullity + rank" Theorem. Rank of a matrix as the dimension of the linar span of its columns and as the dimension of the linear span of its rows.
    Reference: Apostol, Vol. I, Sections 16.10, 16.2, 16.3, 16.6, 16.7. See also the following: notes and exercises
    Assigned exercises: Sections 16.4, 16.8, 16.12, the exercises of the previous notes (except for the inverse matrices). Moreover here are some solved exercises of chapter 14

  • Week 14 (June 4 to june 8):
    Topics: Invertible matrices and inverse matrix. Computation of the inverse matrix with Gaussian elimination. Systems of linear equations: complete description of the set of solutions (Rouche'-Capelli Theorem). Explicit algorithm for computing the rank of a matrix and the solutions of a linear system: Gaussian elimination. Passing from the matrix representing a trasformation with respect to some bases to the matrix representing the same transformation with respect to other bases. Determinant of a n by n matrix: definition and basic properties.
    Reference: Inverse matrices and linear systems: Apostol, Vol. I, Sections 16.17, 16.18 and 16.19, see also the "notes and exercises" of the previous week. Change of basis: Apostol, Vol. II, Section 4.9. Determinants: Apostol, Vol. II, Chapter 3, except 3.10, 3.15 and 3.16.
    Assigned exercises: Linear systems: the "notes and exercise" of the previous week. Moreover Apostol, Vol. I Section 16.20 and 16.21. Determinants: Apostol, Vol. II, Section 3.6 (except 3.6, 3.7, 3.8, 3.9). Section 3.11: n.1 and n.7.

  • Week 15 (June 11 to june 15):
    Topics: Determinant of block-triangular matrices. Inverse matrix and determinants. Cramer's rule. Eigenvalues and eigenvectors: examples in finite dimension and examples in infinite dimension. Eigenspaces. Linear independence of eigenvectors belonging to different eigenvalues. Characteristic polynomial. Expression of the second and of the last coefficient of the characteristic polynomial in function of the trace and of the determinant of the matrix.Diagonalization. Fast calulation of high powers of diagonalizable matrices. Real and complex eigenvalues of a real matrix.
    Reference: Apostol, Vol. II, Sections 4.1, 4.2, 4.3, 4.5, 4.6, 4.9.
    Assigned exercises: Linear systems: the "notes and exercise" of the previous week. Moreover Apostol, Vol. II, Sections 4.4, 4.8, 4.10. and also Exercises

  • Week 16 (June 18 to june 23):
    Topics: Relation between the eigenvalues of a matrix and its trace and determinant. Relation between the dimension of the eigenspace and the multiplicity of the eigenvalue as zero of the characteristic polynomial (the former is lesser than or equal to the latter). Hermitian transformations of euclidean complex linear spaces. Spectral theorem. Complex hermitian and real symmetric matrices. Orthogonal matrices. Orthonormal diagonalization of real symmetric matrices. Real quadratic forms. Reduction of a real quadratic form to diagonal form. Application: signature of a real quadratic form, maximum and minimum of a real quadratic form on the unit sphere, reduction of conics to canonical form.
    Reference: Apostol, Vol. II, Full Chapter 5, except sections 5.17
    Assigned exercises: Section 5.5 except n. 8,9,10. Section 5.11, Section 5.15 and also Exercises june 23