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 :
WED 2.00 - 3.45, room 7,
THU 9.30 - 11.15, room 7,
FRI 2.00 - 3.45, room 7,

Timetable tutoring (Dr. Pietro Sabatino):
FRI 9.30 - 11.15, room C9

Office hours:
By appointment (please send me an e-mail).

Textbook: T. Apostol, Calculus, Vol. I and II.

ANNOUNCEMENTS: NEW! Due to overlapping with other exams, the last winter session has been postponed to Feb 26 (same time and room) (see below)

Syllabus: Chapters 12, 13, 14, 15, 16 of volume I of the textbook and Chapters 3, 4, 5 of volume II. See also pdf

Exams:
The exam consist 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: july 10 at 2.00 p.m., room 11. Oral: WED luly 17, starting at 10.45, room B11.
  • 2nd Session - Summer: july 22 at 11.00 a.m., room B11 (note the change of room).
  • 3rd Session - Autumn: sept 12 at 2.30 p.m., room C7
  • 4rd Session - Autumn: sept 19 at 11.00 a.m., room C7.
  • Extra Session: oct 30 at 11.30 a.m., room 9
  • 5th Session - Winter: feb 10 at 10.30, room B7
  • 6th Session - Winter: feb 26 at 10.30, room B7 NOTE THE CHANGE OF DATE!!!!


Exam results :
  • 2nd session (july 22, 2013). solutions (some solutions have been slightly expanded)
  • 6th session (feb 26, 2014). Admitted to the oral part: Mohamad Abdul Al (12).Please conctact me by e-mail to agree a date for the oral exam. The others are not admitted.


Text and solutions of previous years year written tests:
  • 3rd session 2010-'11 (Corrected mistake in the solution of Exercise 2, thanks to the students who pointed out that).


Weekly description of the topics of the lectures, with assigned exercises:
(Note on exercises extracted from previous exams: even if there is the solution, try to solve the exercise without looking at it. Then compare with the solution)

  • Week 1 (March 4 - 10)
    Topics: V(n) (space of real n-tuples). Addition and scalar multiplication in V(n). Dot product. Norm. Orthogonality. Orthogonal projection of a vector A along a vector B and orthogonal decomposition. Angle. Unit vectors. Linear combinations and linear span of a subset of V(n).
    Reference: Apostol, Calculus Vol. I, Sections 12.1-3, 12.5-7, 12.9-10 and the beginning of 12.12
    Assigned exercises: Apostol, Calculus Vol. I, Sections 12.4, 12.8, 12.11. Section 12.15: Exercises 1 - 4
  • Week 2 (March 11 - 16)
    Topics: Linear independence. How to verify linear independence and ti find maximal independent subsets. Bases of V(n). Components with respect to a giben basis. Orthogonal and orthonormal bases. V(n)(C) (space of complex n-tuples).
    Reference: Apostol, Calculus Vol. I, Sections 12.12-14 and 12.16.
    Assigned exercises: Apostol, Calculus Vol. I Sections 12.15 and 12.17.
    1st Midterm I, 2010-'11: Ex. 1(a) and 2.
    1st Midterm II, 2010-'11:Ex. 1 and 2.
    3rd session 2011-'12: Ex.3.
    5th session 2011-'12: Ex. 1
  • Week 3 (March 18 - 23)
    Topics: Lines. Distance point-line (in V(n). Planes. Determinants of matrices of order two.
    Reference: Apostol, Calculus Vol. I, Sections 13.1-5 and 13.6-7, 12.9-10.
    Assigned exercises: Apostol, Calculus Vol. I, Sections 13.5, 13.8.
    1st midterm 2010-'11, II: Ex. 3.
    4th session 201-'11: Ex. 1.
    6th session 2011-'12: Ex. 1.
  • Week 4 (March 25 - 30)
    No classes
  • Week 5 (Only April 4 and 5)
    Topics: Determinants of order two. Determinants of order three. Cross product in V(3). Determinant as a triple product. Area and volume. Linear independence and determinants. Cramer's rule. Cartesian equations defining line and planes. Normal vectors and cartesian equations. Distance point-plabe in V(3) and point-line in V(2).
    Reference: Apostol, Calculus Vol. I, Sections 13.9-10, 13.12-13, 1315-16.
    Assigned exercises: Apostol, Calculus Vol. I, Sections 13.11, 13.14, 13.17
    1st midterm 2010-'11, I: Ex. 1 (full). Ex. 3.
    1st midterm 2010-'11, II: Ex. 1. Ex. 2. Ex. 3.
    2nd session 2010-'11: Ex. 1.
    3rd session 2010-'11: Ex. 1.
    4th session 2010-'11: Ex. 2.
    1st session 2011-'12: Ex. 1(b).
    3rd session 2011-'12: Ex.2.
    4th session 2011-'12: Ex. 1. Ex.2
    5th session 2011-'12: Ex.1.
    6th Session 2011-'12: Ex. 2.
  • Week 6 (Only April 8 - 13)
    Topics: Conics: eccentricity, focus and directrix. Polar equation. Conics with central symmetry (= ellipses and hyperbola) and their cartesian equation.
    Reference: Apostol, Calculus Vol. I, Sections 13.19-20, 13.22.
    Assigned exercises: Apostol, Calculus Vol. I, Sections 13.21
    1st midterm 2010-'11, I: Ex. 4.
    3rd session 2010-'11: Ex. 2.
    2nd session 2011-'12: Ex. 1.
    3rd session 2011-'12: Ex.3.
    5th session 2011-'12: Ex.2.
  • Week 7 (Only April 15 - 20)
    Topics: Conics with central symmetry (= ellipses and hyperbola) and their cartesian equation (conclusion). Cartesian equations of parabolas. Vector valued fubctions: generalities. Regular curves. Change of parameter. Tangent line. Examples (refelection properties of conics). Velocity vector and speed. Unit tangent and unit normal vectors. Decomposition of the acceleration vector.
    Reference: Apostol, Calculus Vol. I, Sections 13.23. 14.1-3, 14.5-6. 14.8
    Assigned exercises: Apostol, Calculus Vol. I, Sections 13.24-25, 14.7 and 14.9.
    1st midterm 2010-'11, I: Ex. 5.
    1st midterm 2010-'11, II: Ex. 4,5.
    1st session 2010-'11: Ex. 3.
    2nd session 2010-'11: Ex. 2, 3
    3rd session 2010-'11: Ex. 2.
    2nd session 2011-'12: Ex. 2.
    5th session 2011-'12: Ex.3. 6th session 2011-'12: Ex. 3.
  • Week 8 (Only April 23)
    Topics: Length of an arc of curve. Arc-length function. Arc-length reparametrization of a regular curve. Curvature. Characterization of circles as the only plane curves with constant curvature. Other examples. Curves in polar coordinates.
    Reference: Apostol, Calculus Vol. I, Sections 14.10-12 (only the last page of 14-12, where we take the formulas of arc-length as integral of the speed as definition of arc-length), 14.14, 14.16.
    Assigned exercises: Apostol, Calculus Vol. I, Sections 14.13 and 14.15.
  • Week 9 (Only May 2 and 3)
    Topics: Plane curves in polar coordinates. The areal speed of a plane radial motion is constant. Space curves in cylindrical coordinates. Central space motion: it is plane and has constant areal speed (2nd Kepler's law). Central space motion such that the norm of acceleration is the inverse of the distance from the center times a fixed constant: the underlying curve is an ellipse with a focus at the center (1st Kepler's law). Linear spaces. Linear subspace. Examples. Linear combinations. The subspace spanned by a given set.
    Reference: Apostol, Calculus Vol. I, Sections 14.16-18, 14.20 (except 3rd Kepler's law). Sections 15.1-4 and 15.6.
    Assigned exercises: Apostol, Calculus Vol. I, Sections 14.19 and 14.21
  • Week 10 ( May 6 - 11)
    Topics: Linear independence. Finite vs infinite-dimensional linear spaces. Bases of a finite-dimensional linear space. Components with respect to a basis.
    Inner products (real or complex). Examples. Cauchy-Schwartz inequality. Norm. Distance. Orthogonality. Orthogonal and orthonormal bases. Example: space of rigonometric polynomials. Components with respect to orthogonal bases. Computing the inner product with respect to an orthonormal basis.
    Projection on a 1-dimensional linear subspace. Gram-Schmidt orthogonalization. Examples. The orthogonal of a finite-dimensional linear subspace. Orthogonal decomposition theorem. Projection on a finite-dimensional linear subspace.
    Reference: Apostol, Calculus Vol. I,
    Assigned exercises: Apostol, Calculus Vol. I, Sections 15.5, 15.10, 15.13. Section 15.17: EX. 1-5.
    2nd midterm 2010-'11: Ex. 1 and Ex. 5 1st session 2010-'11: Ex. 1 and Ex. 2 2nd session 2010-'11: Ex. 3
    1st session 2011-'12: Ex. 4.
    2nd session 2011-'12: Ex. 4. 3rd session 2011-'12: Ex. 1
    4thsession 2011-'12: Ex.2.
  • Week 11 ( Only May 15 and 16)
    Topics: Orthogonal decomposition theorem and Projection on a finite-dimensional linear subspace: examples from geometry and analysis (best approximation). The orthogonal subspace in the finite-dimensional case. Distance (point)-(linear subspace). Distance (point)-(plane) in V(n).
    Linear transformations: definition and examples. Matrices. Matrix multiplication. The linear map associated to a matrix.
    Reference: Apostol, Calculus Vol. I, Section 15.16.
    Apostol, Calculus Vol. I, Sections 16.1, 16.13, 16.15.
    Assigned exercises: Apostol, Calculus Vol. I, Section 15.17.
    3rd session 2010-'11: Ex. 5
    1st session 2011-'12: Ex. 1.
    2nd session 2011-'12: Ex. 4. 3rd session 2011-'12: Ex. 1
    6thsession 2011-'12: Ex.4.
  • Week 12 ( Only May 23 and 24)
    Topics: The identity matrix. Null-space, range and rank of a linear transformation. Nullity + rank theorem. Rank of a matrix.Systems of linear equations: the Rouche'-Capelli theorem. Injectivity and surjectivity of a linear transformation.
    Reference: Apostol, Calculus Vol. I, Sections 16.2, 16.3, 16.17.16.18.
    Supplementary notes and exercises
    Assigned exercises: Apostol, Calculus Vol. I, Section 16.4, 16.16, 16.20: Ex. 1-10
    2nd session 2011-'12: Ex. 3.
  • Week 13 ( Only May 27 - 31)
    Topics: Bijective (invertible) linear transformations. Correspondence {linear transformations from V(n) to V(m)} - {matrices with m rows and n columns}. Correspondence between matrix multiplication and composition of linear transformations. Invertible matrices and their rank. Calculation of the inverse matrix witj gaussian elimination.
    Linear trasformations with prescribed values. The matrix representing a linear transformation with respect to a basis of the domain and a basis of the target space. Examples: projections, reflections, rotations of the space around a line passing trough the origin. Change-of-basis matrices. Formula relating matrices representing the same linear transformation with respect to different bases.
    Reference: Apostol, Calculus Vol. I, Sections 16.17, 16.18. 16.19, 16.9, 16.10..
    Supplementary notes and exercises (to appear)
    Assigned exercises: Apostol, Calculus Vol. I, Section 16.8, 16.12, 16.20: Ex. 11-16, 16.21.
    2nd midterm 2010-'11: Ex. 2.
    3rd session 2010-'11: Ex. 4(a)(b)
    1st session 2011-'12: Ex.3
    2nd session 2011-'12: Ex. 3
    3rd session 2011-'12: Ex. 4
    4th session 2011-'12: Ex. 4
    5th session 2011-'12: Ex. 4
  • Week 14 ( June 3 - 8)
    Topics: Examples of representative matrices and applications.
    Multilinear alternating row functions. Calculation rules. Uniqueness of m.a.r. functions up to multiplicative constant. The determinant. Calculation of determinants with row elimination. Determinants and rank of a matrix. Laplace expansions. Determinant of a product. Determinant and inverse matrix. Cramer's rule.
    Eigenvalues and eigenvectors: definitions and examples. Independence of eigenvectors of distinct eigenvalues. Eigenvalues as zeroes of the characteristic polynomial.
    Reference: Apostol, Calculus Vol. II, Chapter 3.
    Calculus, Vol. II, Sections 4.1-3, 4.5, 4.6
    Moreover: Supplementary notes and exercises on linear transformation and matrices, II .
    Assigned exercises: Apostol, Calculus Vol. II, Sections 3.6, 3.11, 3.17 and Sections 4.4. Section 4.8 Ex. 1 - 11.
  • Week 15 ( June 10 - 16)
    Topics: Real and complex eigenvalues. Diagonalizibility and diagonalization of a linear transformation and of a matrix. Dimension of the eigenspace versus multiplicity of the eigenvalue as zero ofthe characteristic polynomial. Trace, determinants and eigenvalues.
    Hermitian, skew-hermitian, symmetric and skew-symmetric linear transformations, and their representative matrices with respect to an ORTHONORMAL basis. Spectral theorem. Diagonalization with an orthonormal basis. Orthogonal matrices. Diagonalization with an orthogonal matrix. Real quadratic forms and their matrices.
    Reference: Apostol, Calculus Vol. II, Chapter 3.
    Calculus, Vol. II, Sections 4.7, 4.9.
    Vol. II Sections 5.1-5.4, 5.6-5.10, 5.12
    Assigned exercises: Apostol, Calculus Vol. II, Sections 4.10, 5.5 (up to Ex. 7), 5.11.
    Moreover: Exercises june 15
    Moreover: 2nd midterm 2010-'11: Ex. 3
    1st session 2010-'11: Ex. 4
    2nd session 2010-'11: Ex. 4
    3rd session 2010-'11: Ex. 3 and Ex. 4
    4th session 2010-'11: Ex. 4
    1st session 2011-'12: Ex. 3
    2nd session 2011-'12: Ex. 5
    3rd session 2011-'12: Ex. 5
    5th session 2011-'12: Ex. 5
  • Week 16 ( June 17 -22)
    Topics: Real quadratic forms and their matrices (continuation). Canonical diagonal form of a real quadratic form. Sign of a real quadratic form (positive, semipositive, negative, seminegative, indefinite). Maximum and minimum of the restriction of a req quqdratic form to the unit sphere. Conics: reduction to canonical form.
    Reference: Apostol, Calculus Vol. II, Chapter 3.
    Calculus, Vol. II, 5.12, 5.13, 5.14, 5.16, 5.18
    Assigned exercises: Apostol, Calculus Vol. II, Section 5.15
    Moreover: Exercises june 22
    Moreover: 2nd midterm 2010-'11: Ex. 4
    1st session 2010-'11: Ex. 5
    2nd session 2010-'11: Ex. 5
    4th session 2010-'11: Ex. 5
    1st session 2011-'12: Ex. 5
    4th session 2011-'12: Ex. 5
    5th session 2011-'12: Ex. 5