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 2013-'14, ENGINEERING SCIENCES

Instructor: Prof. Giuseppe Pareschi

Teaching Assistant: Dr. Ha Tran Nguyen Thanh

Timetable course :
WED 11.30 - 1.15 pm and 2.00 - 2.45 (upon students' request, the last hour can be moved up, skipping the lunch-break)
THU 10.30 - 11.15
FRI 2.00 - 3.45
All classes will take place in room 7.

Timetable tutoring (Dr. Ha Tran Nguyen Thanh):
FRI 9.30 - 11.15 room 7

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

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

ANNOUNCEMENTS: NEW!
  • The next written exams will be on feb 11 and feb 18 at 2.00 pm. Room C4.
    • During the written exams it will be possible to consult notes and/or books.


    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.
      Reference: Apostol, Calculus Vol. I, Sections from 12.1 to 12.6.
      Assigned exercises: Apostol, Calculus Vol. I, Section 12.4 (all) and Section 12.8, from n.1 to n. 11.
    • Week 2 (March 7 - 13)
      Topics: Projection. Given vectors A and B how to construct vectors of the form xA+yB forming a prescribed angle with A.
      Linear span of a finite collection of vectors. Discussion of various particular cases. Linear independence.
      Reference: Apostol, Calculus Vol. I, Sections 12.9. 12.10, 12.12
      Assigned exercises: Apostol, Calculus Vol. I, Section 12.8, from n.1 from n.12 to the end. Sections 12.11 and 12.15 (all). n. 11.
    • Week 3 (March 14 - 19)
      Topics: LInear independence and dependence (continuation). 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 12.13, 12.14.
      Assigned exercises: Apostol, Calculus Vol. I, Section 12.15.
      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 4 (March 21 - 26)
      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.
      Reference: Apostol, Calculus Vol. I, Sections 13.1-4. 13.6-7. Note that in class it was explained how to compute the distance between a point and a line in V(n), for any n, while in the book this case is treated only in the case n=2.
      Assigned exercises: Apostol, Calculus Vol. I, Sections 13.5 and 13.8.
      More exercises (2nd series)
      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
      3rd session 2012-'13. Ex.1
    • Week 5 (March 27 - April 3)
      Topics: Determinants of order 3 (continuation). 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. Normal vector and cartesian equation of a plane in V(3). Distance point-plane in V(3). Volume. 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 13.9-10, 13.12-13 and 13.15-16. Note that in class it was given a full treatment of determinants of order 3, while in the book there is only the Laplace expansion from the first row (actually, this is taken as definition).
      Assigned exercises: Apostol, Calculus Vol. I, Sections 13.11, 13.14 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 6 (April 4 - April 10)
      Topics: Conic sections: polar equation, cartesian equation, conics with central symmetry.
      Vector valued functions: rules of calculations. Reparametrizations. Regular curves. Velocity and accelaration vectors, speed, tangent line, unit tangent vector, unit normal vector, formula for the acceleration. Reference: Apostol, Calculus Vol. I, Sections 13.19-20 and 13.22-23. Sections 14.1-3, 14.5-6, 14.8
      Assigned exercises: Apostol, Calculus Vol. I, Sections 13.21, 13.24, 13.24, 13.25, 14.4, 14.7, 14.9. Note that, concerning vector valued functions, some of the following exercises, contained in previous years' exams, are very similar, if not equal, to some exercises of the book.
      First midterm I, 2010-'11, Ex.4 and 5.
      First midterm II, EX. 4 and 5.
      1st session 2010-'11. Ex.3
      2nd session 2010-'11, Ex.2 and 3(a)
      3rd session 2010-'11, Ex.2
      1st session 2011-'12, Ex. 2(a)(b)
      2nd session 2011-'12, Ex.1 and 2
      4th session 2011-'12. Ex. 3.
      5th session 2011-'12. Ex. 2 and 3. 6th session 2011-'12. Ex 3(a)
      1st session 2012-'13. Ex. 3
      2nd session 2012-'13, Ex. 2
      3rd session 2012-'13, Ex. 2
      4th session 2012-'13. Ex. 2
      Special session 2012-'13, Ex.2
      5th 2012-'13. Ex.2
      6th session 2012-'13. Ex.2
    • Week 7 (April 11 - 17)
      Topics: Length of an arc of curve and arc-length function. Curvature. Velocity, speed, acceleration, curvature in polar coordinates. Area-speed. Cylindrical coordinates. Curves with radial acceleration. Kepler's laws (only the first two).
      Reference: Apostol, Calculus Vol. I, Sections 14.10-12 (note: you can skip most of this sections, and take as definition of arc-length Theorem 14.13 and the subsequent formula (immediately before Example 1. Then you should study Examples 1 and 2.) Sections 14.14, 14.16-18, 14.20.
      Assigned exercises: Apostol, Calculus Vol. I, Sections 14.13, 14.15, 14.19 and 14.21 . Here are some solutions
      Moreover you should study all the remaining exercises on vector valued functions in the previous years' exams. Most, if not all, of them are equal or completely similar to exercises in the textbook.
    • Week 8 (April 30 - May 7,8,9)
      Topics: Linear spaces (real or complex). Linear subspaces. Examples. Linear dependence and independence. Linear span. Finite-dimensional vs infinite-dimensional linear spaces. Bases and dimension of finite-dimensional linear spaces. Inner products and norms (in the real and in the complex case).
      Reference: Apostol, Calculus Vol. I, Sections 15.1-4, 15.6-8, 15.10
      Assigned exercises: Apostol, Calculus Vol. I, Sections 15.5, 15.9, 15.12.
    • Week 9 (May 14 - 21)
      Topics: Orthogonality, orthogonal and orthonormal bases, Gram-Schmidt orthogonlaization. The orthogonal subspace. Orthogonal decomposition theorem. Orthogonal projection (onto a finite-dimensional subspace). Distance from a finite dimensional subspace. Interpretation as best approximation. Examples from analyis: trigonometric polynomials and Legendre polynomials.
      Reference: Apostol, Calculus Vol. I, Sections 15.11, 15.13-15.
      Assigned exercises: Apostol, Calculus Vol. I, Sections 15.12, 15.16. Moreover:
      Second midterm, 2010-'11. Ex. 1
      First session 2010-'11. Ex. 1
      Fourth session 2010-'11. Ex. 3
      First session 2011-'12. Ex.4
      Second session 2011-'12. Ex.4
      Third session 2011-'12. Ex.1
      6th session 2011-'12: Ex.4
      3rd session 2012-'13: Ex.3
      Special session 2012-'13: Ex. 4
      5th session 2012-'13: Ex.4
    • Week 10 (May 22 - May 29)
      Topics: Linear trasformations. Matrices. Matrix multiplication. Matrical notation for linear systems. Sum and scalar multiplication of matrices. Properties of matrix multiplication. Identity matrices. The linear transformation associtaed to a matrix.
      Null-space and range of a linear trasformation. Nullity+rank theorem. Interpretation of null space and range for the transformation associated to a matrix. Rank of a matrix as the maximal number of independent columns and of independent rows. Rouche'-capelli Theorem. Reference: Apostol, Calculus Vol. I, Sections 16.1-3, 16.10, 16.13, 16.15, 16.17, 16.18
      Assigned exercises: Apostol, Calculus Vol. I, Sections 16.4, 16.16, 16.20
      Moreover, there are these Supplemntary notes and exercises
    • Week 11 (May 30 - june 6)
      Topics: Null-space and injectivity. Injective, surjective and bijective linear transformations.
      Invertible matrices and inverse matrices. Invertibility and rank. Calculation of the inverse matrix by means of gaussian elimination.
      Linear tranformations with precribed values on a vector of a given basis.
      Matrix representing a given linear transformation with respect to given bases of the target and of the source. Examples: projections and reflections.
      Change of basis.
      Reference and Assigned exercises: Supplementary notes and exercises II . or Apostol, Vol. I: Sections 16.7, 16.9, 16.10, 16.13, 16.14, 16.19
      Exercises an the textbook: Apostol, Calculus Vol. I, Sections 16.8, 16.12, 16.16 16.20
      Moreover:
      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 12 (june 9 - june 13)
      Topics: Determinants (summary without many proofs): properties, calculation by means of gaussian elimination. Laplace expansions. Determinant of a product. Determinant and linear independence. Determinant and inverse matrix. Cramer's formula.
      Eigenvalues and Eigenvectors of linear transformations and of matrices. . Definition. Examples. Linear independence of eigenvectors of distinct eigenvalues. Characteristic polynomial. Trace and determinant. Eigenvalues as roots of the characteristic polynomial. Diagonalizability and diagonalization of linear transformations and matrices.
      Reference Determinants: Apostol, Vol. II, Chapter 3
      Eigenvalues and eigenvectors: Apostol, Vol II, Chapter 4
      Assigned exercises The sections of exercises of Chapters 3 and 4. Specifically, for determinants: Sections 3.6, 3.11, 3.17. For eigenvalues: Sections 4.4, Examples 1,2 and 3 of Section 4.6, Section 4.8, section 4.10.
      Moreover here you can find more exercises
    • Week 13 (june 15 - june 20)
      Topics: Summary about reasons why a linear transformation could not be diagonalizable. Example: rotations of V(2).
      Review of complex euclidean spaces. Hermitian transformations. Eigenvalues and eigenvectors of hermitian transformations. Hermitian and symmetric matrices. Theorem: a transformation is hermitian if and only if the matrix representing it with respect to an ORTHONORMAL basis is hermitian.
      Sperctral theorem for real symmetric matrices. Orthogonal matrices.
      Real quadratic forms. Matrix of a real quadratic form. Change of basis for real quadratic forms. Diagonalization of real quadratic form via an orthonormal basis of eigenvectors of the matrix: canonical form. Application: sign of a real quadratic form (positive, semi-positive, negative, semi-negative, indefinite). Study of the conic defined by a polynomial of degree 2 in two variables. Reference Apostol, Vol. II, Chapter 5 (But skip the examples given by differential operators and the parts on skew-hwrmitian transformations). Sections 5.1-4, 5.6-10, 5.11-14
      Assigned exercises Sections 5.11 and 5.15
      More exercises (21/6)
      Exercises in previous exams:
      2nd midterm, 2010-'11: Ex. 3 and 4.
      1st session 2010-'11: Ex. 4 and 5
      2nd session 2010-'11: Ex. 5
      3rd session 2010-'11: Ex. 4
      4th session 2010-'11: Ex. 4(a) and 5(a)
      1st session 2011-'12: Ex. 3 and 5
      3rd session 2011-'12: Ex.5
      4th session 2011-'12: Ex. 5
      5th session 2011-'12: Ex.5
      6th session 2011-'12: Ex. 5(a)(b)
      1st session 2012-'13: Ex. 5(a)
      2nd session 2012-'13: Ex.5
      3rd session 2012-'13: Ex.4
      4th session 2012-'13: Ex.4
      special session 2012'-'13: Ex. 5
      5th session 2012-'13: Ex. 5
      6th session 2012-'13: Ex. 5
    • Week 13 (june 15 - june 20)
      Exercises (in one exercise it was explained that the minimal (respectively maximal) eigenvalue of a symmetric matrix is the minimum (respectively maximum) on the unit sphere of the corresponding quadratic form (see Apostol, Vol. II, Section 5.16).


    Intermediate tests
    • First test (march 14).
      Text and solutions
      The following students got one point: Brigato L., Cicchetti L., Ferraro M., Galieti R., Marcucci G., Moulton J. Pretagostini F.,
    • Second test (march 28).
      The following students got one point: Brigato L., Cicchetti L., Ferraro.
      The following students got half point:Kaur Jasminder, Galieti R., Marcucci G., Mahnoor Malik, Moulton J. Pretagostini F., Valentini C.
    • Third test (april 4).
      Text and solution
      Grades: Brigato L.: 1,0; Cicchetti L.: 0,8, Ferraro, M.: 1,2; Graziano G.: 0,8; Kaur Jasminder: 1,2; Galieti R.: 1,0; Leonova K.: 0,6; Mahnoor Malik.: 0.8; Pretagostini F.: 0,8, Valentini C. 1,0
    • Fourth test (april 11).
      Text and solution
      Grades: Cicchetti L.: 0,3: , Ferraro, M: 0,7; Graziano G.: 0,2; Kaur Jasminder: 0,5; Galieti L.: 0,2; Marcucci G. 1,0; Pretagostini F.: 0,5, Valentini C. 0,7.
      The others didn't get points.
    • Fifth test (may 9).
      Solution
      Grades: Brigato: 0,5. Graziano: 0.5, Pretagostini: 0,7. Malik: 0,5. Marcucci: 0,7. Kaur: o,1., Cicchetti: 0,7, Valentini: 0,6, Ferraro: 0,6, Galieti: 0,6.


    • Sixth test (may 23).
      Solution
      The following students got a positive grade: Brigato: 0.2; Graziano 0,2; Kam Jasminder: 0,2, Pretagostini: 0,7; Mahnoor Malik: 0,5; Marcucci: 1,0; Volentini: 0,3; Ferraro: 0,2; Galieti: 0,9


    • Seventh test (june 6).
      Solution
      Brigato: 0,6. Cicchetti: 0,4. Kaur: 0,4. Galieti: 0,6. Ferraro: 0,6. Malik: 0,2. Graziano: 0,1. Marcucci: 0,9. Pretagostini: 0,5. Valentini: 0,1.


    • 8th test (june 25).
      Solution
      The following students got a positive grade: Brigato: 0.1; Kaur: 0,5, Pretagostini: 0,7; Marcucci: 0.8; Volentini: 0,1; Ferraro: 0,7; Galieti: 0,9, Graziano 0,6, Valentini: 0.7, Leonova: 0,1, Cicchetti: 1,1, Mancuso: 0,5.


    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 a course, we will 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 Calendar :
    • Next written exams:
      february 11, at 2.00 pm, room C4
      february 18, at 2.00 pm, room C4.
      The dates of the oral parts to be announced.


    Exam results :

    • Third session (12-09) and fourth session (24-09): no students admitted to the oral part. Text and solutions of bothe written sessions will be posted as soon as possible.
    • Fifth session (feb 11, 2015): no students came. Sixth session (feb 18): no students admitted to the oral part. Text and solution


    Text and solutions of previous years year written tests: