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 2014-'15, ENGINEERING SCIENCES

Instructor: Prof. Giuseppe Pareschi

Teaching Assistant: Dr. Andrea Del Monaco

Timetable course :
WED 2.00 - 3.45 pm
THU 9.30 - 11.15 am
FRI 2.00 - 3.45 pm
All classes will take place in room 7.

Timetable tutoring (Dr. Andrea Del Monaco):
WED 11.30 - 13.15 room 7

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

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

ANNOUNCEMENTS:
(1) NEW! NEXT WRITTEN EXAMS:
MARCH 30, 4.00 pm, room 1

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

Weekly description of the topics of the lectures, with assigned exercises:
  • Week 1 (March 3 - 6)
    Topics: V(n) (space of real n-tuples). Addition and scalar multiplication in V(n). Dot product. Norm. Angle between two vectors. Unit vectors of the coordinate axes.
    Reference: Apostol, Calculus Vol. I, Sections from 12.1 to 12.10.
    Assigned exercises: Apostol, Calculus Vol. I, Section 12.4, 12.8 and 12.11
  • Week 2 (March 9 - 13)
    Topics: Linear span of a finite collection of vectors. Discussion of various particular cases. Linear independence. Bases of V(n). Components of a vector with respect to a given basis. Orthogonal and orthonormal bases. Examples of solution of linear systems with gaussian elimination.

    Reference: Apostol, Calculus Vol. I, Sections 12.12, 12.13, 12,14,
    Assigned exercises: Apostol, Calculus Vol. I, Section 12.15 n. 11.
    Moreover:
    First midterm I, 2010-'11, Ex. 1(a), Ex. 2
    First midterm II, 2010-'11, Ex.2.
    3rd session 2011-'12, Ex.3.
    Special session 2012-'13, Ex. 1(a).
    More exercises
  • Week 3 (March 13 - 20)
    Topics: Lines in V(n). Distance point-line in V(n). Normal vectors and cartesian equations of lines in V(2). Distance point-line in V(2). Planes in V(n). Determinants of order 2 and 3. Definition, Laplace expansions. First properties. Cross product of two vectors in V(3). Area of a parallelogram in V(3). Determinant of a matrix of order 3 as a mixed product.
    Reference: Apostol, Calculus Vol. I, Sections 13.1-4. 13.6-7, 13.9-10, 13.12.
    Assigned exercises: Apostol, Calculus Vol. I, Sections 13.5, 13.8. 13.11, 13.14 (up to Ex. 16)
    More exercises (2nd series) (except the last exercise)
    First midterm II, 2010-'11, Ex.3.
    4th session 2011-'12, Ex.1.
    6th session 2011-'12. Ex 1(a)
    1st session 2012-'13. Ex. 1
    2nd session 2012-'13. Ex. 1
    2nd session 2013-'14, Ex.1 and Ex.2
  • Week 4 (only march 25 and 26)
    Topics: Normal vector and cartesian equation of a plane in V(3). Distance point-plane in V(3). Linear independence of 3 vectors in V(3) and determinants. Application to linear systems of 3 inequations in 3 unknowns. Cramer's rule.
    Reference: Apostol, Calculus Vol. I, Sections 13.13 and 13.15-16.
    Assigned exercises: Apostol, Calculus Vol. I, Sections 13.14 (Ex. from 17 to 20)and 13.17
    More exercises (3rd series)
    First midterm I, 2010-'11, Ex.3.
    2nd session 2010-'11, Ex.1
    4th session 2010-'11, Ex.1.
    3rd session 2010-'11. Ex.1
    4th session 2010-'11. Ex.2
    3rd session 2011-'12, Ex. 2.
    4th session 2011-'12. Ex. 1 and 2.
    6th session 2011-'12. Ex 1 and 2.
    1st session 2012-'13. Ex. 1
    4th session 2012-'13. Ex. 1
    5th 2012-'13. Ex.1
  • Week 5 (only april 1)
    Topics: Conic sections: eccentricty and polar equation. Reference: Apostol, Calculus Vol. I, Sections 13.18-20.
    Assigned exercises: Apostol, Calculus Vol. I, Section 13.21
  • Week 6 (only april 9 and 10)
    Topics: Conic sections: polar equation (cont.). Conic sections with central symmetry.
    Reference: Apostol, Calculus Vol. I, Sections 13.22.
  • Week 7 ( april 14-17)
    Topics: Conic sections with central symmetry (cont.). Standard equations for conic sections.
    Vector valued functions: extending the main rule of differential calculus to them.
    Reference: Apostol, calculus, Vol. I, Sections 13.22-23 and 14.1-3. Assigned exercises: Apostol, Calculus Vol. I, Sections 13.24 and 13.25. Section 14.4.
    First midterm I, 2010-'11, Ex.4.
    2nd session 2010-'11, Ex.2
    3rd session 2010-'11, Ex.2
    2nd session 2011-'12: Ex.1
    3rd session 2011-'12, Ex. 2.
    5th session 2011-'12. Ex. 2.
    2nd session 2012-'13. Ex. 2
    4th session 2012-'13. Ex. 2
    1st session 2013-'14 Ex. 2
  • Week 8 ( april 21- 24)
    Topics: The underlying curve of a vector valued funcion. Examples. Reparametrizations. Velocity vector and tangent line. Example. Reflection properties of conic sections. Acceleration vector, examples. Relation between acceleration and velocity. Unit tangent vector and unit normal vector. Normal line. Osculating plane. Angle of inclination and normal vector of a plane curve. Reference: Apostol, Calculus Vol. I, Sections 14.5-6 and 14.8.
    Assigned exercises: Apostol, Calculus Vol. I, 14.7 and 14.9
    First midterm I, 2010-'11, Ex.5.
    First midterm, II, 2010-'11, Ex. 4 and 5
    1st session 2010-'11, Ex.3
    2nd session 2010-'11: Ex. 3(a)
    1st session 2011-'12, Ex. 2.
    2nd session 2011-'12. Ex. 2.
    4th session 2011-'12. Ex 3.
    5th session 2011-'12. Ex 3.
    6th session 2011-'12. Ex 3.
    1st session 2012-'13. Ex. 3
    3th session 2012-'13. Ex. 3
    Special session, 2012-'13 Ex.2
  • Week 9 (only april 29 and 30)
    Topics: Arc-length. Curvature. Theorem: a regular plane curve with non-zero constant curvature is (a part of) a circle. Reference: Apostol, Calculus Vol. I, Arc-length: only the final definitions (Statement of Theorem 14.13 and p.534)
    Section 14.14.
    Assigned exercises: Apostol, Calculus Vol. I, Sections 14.13 and 14.15
  • Week 10 (may 5- 8)
    Topics: Calculations with polar coordinates. Central motion: the position vector sweeps out area at constant rate. Geometric proof of Kepler's laws. Reference: Apostol, Calculus Vol. I, Sections 14.16-18 and 14.20
    Assigned exercises: Apostol, Calculus Vol. I, Sections 14.19, 14.21. Some solved exercises
    You should also look at the exercises on vector valued functions in the past exams
  • Week 11 (may 11-15)
    Topics: Linear spaces. Examples. Linear subspaces. Linear combinations and linear span. Linear independence. Finite-dimensional and infinite-dimensional linear spaces. Bases of a finite-dimensional linear space. Inner products. Real euclidean spaces. Schwartz inequality and triangle inequality. Norm, distance, angles, orthogonality. Orthogonal projection. Examples of inner products. Orthogonal bases. Components with respect to orthogonal bases. Reference: Apostol, Calculus Vol. I, Sections 15.1-5, 15.6-8, and 15.10-11 (except for the example at p. 565).
    Assigned exercises: Apostol, Calculus Vol. I, Sections 14.5, 15.9, 15.12.
  • Week 12 (may 18-22)
    Topics: Gram-Schmidt orthogonalization. Examples: trigonometric polynomials, Legendre polynomials. Orthogonal projection onto a finite-dimensional linear subspace. Orthogonal decomposition theorem. Distance vector-subspace: best approximation.
    Reference: Apostol, Calculus Vol. I, Example in Section 15.12. Sections 15.14-16.
    Assigned exercises: Apostol, Calculus Vol. I, Sections 15.13, 15.17. All exercises of the previous exams concerning the topics of Chapter 15 (bases and dimension, inner producs, orthogonalization, orthogonal decomposition, distance vector-subspace...)
  • Week 13 (may 24-29)
    Topics: Matrix multiplication. Linear spaces of matrices. Matrix notation for linear systems. Linear transformations. The linear transformation from V(n) to V(m) associated to a mxn matrix. Correspondence between matrix multiplication and composition of linear transformations. Linear transformations with prescribed values (on the vectors of a given basis of the domain).
    Null-space and range, nullity and rank of a linear transformation. Null-space and range, nullity and rank of a matrix. The nullity + rank theorem. Consequence: the maximal numer of independent columns of a matrix equals the maximal number of independent rows. Reference: Apostol, Calculus Vol. I, Section 16.15 (matrix multiplication), Sections 16.17-18 (linear systems). Sections 16.1-3 and 16.5-6 (Linear transformations, nullity, rank, composition ...).
    Assigned exercises: Apostol, Calculus Vol. I, Sections 16.4, 16.12 (Only 1.2.3.4). 16.16, 16.21
  • Week 14 (only june 4 and 5)
    Topics: NUll-space and one-t-one linear transformations. The preimage of an element of the target space via a linear transformation. Linear systems: Theorem of Rouche'-Capelli. Invertible linear transformations. Invertible matrices. Invertible matrices have maximal rank (that is, they are non-singular, in the textbook's terminology). Calculation of the inverse matrix. Matrix representing a linear transformation wuth respect to two bases (a basis of the source space and a basis of the target space). Reference: Apostol, Calculus Vol. I, ll the remaining sections of the chapter, except for Section 16.11 (don't study that). A good part of the material of Chaper 16 of the textbook, which is sometimes unclear, can be replaced by the following Supplementary notes and exercises, I and Supplementary notes and exercise, II
    Assigned exercises: Apostol, Calculus Vol. I, Sections 16.4, 16.8, 16.12, 16.20, 16.21 and
    Second midterm, 2010-'11. Ex. 2 First session 2011-'12. Ex.3 Second session 2011-'12. Ex.3 Third session 2011-'12. Ex.4 Fourth session 2011-'12. Ex.4 5th session 2011-'12: Ex.4 1st session 2012-'123: Ex.2 2nd session 2012-'13: Ex.2 3rd session 2012-'13: Ex.4 Special session 2012-'13: Ex. 3
  • Week 15 ( june 8-12)
    Topics: Determinants: only a short summary. Essentially I skipped the chapter on determinants of arbitrary order (Chap. 3 Vol. II). However I gav a definition of determinants (with Laplace expansions) and pointed out some basic properties.
    Eigenvalues, eigenvectors and diagonalization. Reference: Apostol, Calculus Vol. II, Chapter 4
    Assigned exercises: Apostol, Calculus Vol. II, Sections 4.4, 4.8 and 4.10 and More exercises Concerning determinants, only 2.1 (hint: use Binet's theorem: det(CD)=det(C)det(D))
  • Week 16 ( only june 18-19)
    Topics: Symmetric linear transformations. Symmetric matrices. Spectral theorem. Orthogonal matrices. Diagonalization of a symmetric linear transformation with an orthogonal matrix. Quadratic forms. Reduction of a quadratic form to metric canonical form. Reference: Apostol, Calculus Vol. II, Sections 5.2-4, 5.6-10, 5.12-13. Avoid the complex part, and consider only the real case (symmetric linear transformations, orthogonal matrices..).
    Assigned exercises: Apostol, Calculus Vol. II, Sections 5.5, 5.11. Moreover More exercises
    Moreover all exercises of previous exams and ests on symmetric matrices, symmetric linear transformations, quadratic forms
Exams:
The exam consists of a written part and an oral part. We will have:
  • one session in the week after the end of the course;
  • another session in july;
  • two sessions in september;
  • two sessions is february 2015.
Moreover, during the course, we plan to have eight little intermediate tests (each every two weeks). If a student takes one of them and passes it, he/she will get 1/30 to be added to the score of the final exam. In this way a student can add up to 8/30 to the score of the final exam. The tests will take place either during the classes or during the turoring. They will be announced a week before. The first intermediate test will be on friday, march 14, during the class.

Exams results:

First written exam (6-30-'15): Text and solution (only one version)


Second written exam (7-23-'15): Text and solution <

Third written exam: text and solution

Fourth written exam: no students admitted to the oral exam

Extra exam (11-27-2015): text and solution (some calculations in exercise 2 corrected)
No students admitted to the oral

Intermediate tests
  • First test (3/18). Solution (only one version. The other two were completely similar).
  • Second test (4/1). Solution (only one version. The other is completely similar).
  • Test 3 (april 22) Solution (as usual, only for one version)
  • Test 4 (may 13) Text and solution Only one version, as usual
    Grades:
    CAPUANO ELENA: 0.6 , CARBONARI FRANCESCO: 0.9 , COSTA MATTEO: 0.2 , DAMIAN MIRCEA 0.5, DE MATTEIS ANDREA: 1.3 , DI FEDERICO GAIA: 0.1 , GOLAN MEHEDI HOSSAIN: 0.1 , LONGOBARDI FRANCESCA: 0.7 , MALIK MAHRUKH: 0.3 , SEU MATTIA: 0.3 , SURIANI IRENE: 0.7 , ZARRELLI ALEANDRA: 0.9

  • Test 5 Text and solution Only one version, as usual
    Grades:
    CAPUANO ELENA: 0.2 , CARBONARI FRANCESCO: 1.0 , COSTA MATTEO: 1.0 , DAMIAN MIRCEA 1.0, DE MATTEIS ANDREA: 1.0 , LONGOBARDI FRANCESCA: 1.0 , MALIK MAHRUKH: 0.7 , SEU MATTIA: 0.4 , SOUFRI YASSINE: 1.0 , ZARRELLI ALEANDRA: 1.0
    DI FEDERICO and HOSSAIN didn't get a score.
  • Test 6 Text and solution Only one version, as usual
    Grades:
    CAPUANO ELENA: 0.2 , CARBONARI FRANCESCO: 1.5 , COSTA MATTEO: 1.0 , DAMIAN MIRCEA 1.5, DE MATTEIS ANDREA: 1.4 , LONGOBARDI FRANCESCA: 1.2 , MALIK MAHRUKH: 0.7 , SEU MATTIA: 1.0 , SOUFRI Y.: 0.6 , ZARRELLI ALEANDRA: 0.9
  • Tests bonus:
    AMADIO: 0.3
    BEHZAD: 0.5
    CAPUANO: 2.5
    CARBONARI: 5.3
    COSTA: 4.6
    DAMIAN: 4.0
    DE MATTEIS: 5.9
    DI FEDERICO: 0.5
    GIOLAM MEHEDI HUSSEIN: 0.1
    LONGOBARDI: 4.4
    MALIK MAHRUKH: 1.0
    SEU: 4.0
    SOUFRI: 2.7
    SURIANI: 2.5
    TARANTINI: 1.0
    WALEED: 1.0
    ZARRELLI: 5.0

Exams Calendar : Extra exam: the written part will be on nov 27 at 2.00 pm in room C10.

Text and solutions of previous years year written tests: