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 11.30 am - 1.15 pm
THU 9.30 - 11.15 am
FRI 11.30 - 1.15 am
All classes will take place in room 7.
Office hours: tuesday 2.00 pm - 3.00 pm or by appointment. In the instructor's room: 0212 SOGENE building.

Timetable tutoring (Dr. Andrea Del Monaco):
THU 4.00 - 6.00 pm room 7

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

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

ANNOUNCEMENTS:
(1) NEW! EXTRA EXAM. Monday march 27 at 11 am, room C8.
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 1-4)
    Topics: V(n) (space of real n-tuples). Addition and scalar multiplication in V(n). Dot product. Norm. Orthogonality. Orthogonal projection.
    Reference: Apostol, Calculus Vol. I, Sections from 12.1 to 12.9
    Assigned exercises: Apostol, Calculus Vol. I, Section 12.4, 12.8. Section 12.11: all except 3,4,5,6,7,8,9.
  • Week 2 (March 7-11)
    Topics: V(n) (space of real n-tuples). Angle between two vectors. Unit vectors. Linear combinations. Linear span. Linear independence and dependence. Bases of V(n). Examples of solutions of systems of linear equation with gaussian elimination.
    Reference: Apostol, Calculus Vol. I, Sections from 12.10 to 12.15
    Assigned exercises: Apostol, Calculus Vol. I, Sections 12.11, 12.15. More exercises
    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).
    1st test, 2013-'14
    1st test, 2014-15
  • Week 3 (March 13 - 19)
    Topics: V(n)(C) (space of complex n-tuples). Theorem: every independent subset of V(n) is contained in a basis of V(n). Orthogonal non-zero vectors are independent. Orthogonal and orthonormal bases. Components of a vector with respect to a basis. Components of a vector with respect to an orthogonal basis.
    Lines in V(n) and their properties.
    Reference: Apostol, Calculus Vol. I, Sections 13.1-5.
    Assigned exercises: Apostol, Calculus Vol. I, Section 12.16. Section 13.5.
  • Week 4 (only March 23)
    Topics: Cartesian equations for lines in V(n). Normal vector to a line in V(2) and cartesian equation. Distance point-line in V(2).
    Reference: Apostol, Calculus Vol. I, Section 13.4
    Assigned exercises: More exercises 2
    Exercises concerning lines and distane point-line in V(2) in previous exams and tests.
  • Week 5 (only March 30-31)
    Topics: Exercises on distance and lines. Determinants of order two. Transpose matrix. Determinants of order 3: definition. Laplace expansions, various properties. Cross product and area of a parallelogram. Scalar triple product and volume of a parallelepiped. Theorem: 3 vetcors in V(3) are linearly independent if and only if the matrix whose rows (or columns) are the three vectors has non-zero determinant.
    Reference: Apostol, Calculus Vol. I, Sections 13.9-12
    Assigned exercises: Apostol: Section 13.11 More exercise (3rd series): Exercises 3, 5, 6, 7, 13, 14, 15
    Exercises concerning determinants, areas, volumes, in previous years' exams.
  • Week 6 (April 4 - 8)
    Topics: Cramer's rule. Planes: parametric equation and its non-uniqueness. Plane through three non-collinear points. Other elementary properties of planes. Planes in V(3): normal vector, cartesian equation. Distance point-plane in V(3). Point of a plane which is closest to a given point. Lines in V(3): skew, incident, coplanar.
    Reference: Apostol, Calculus Vol. I, Sections 13.13-16
    Assigned exercises: Apostol: Sections 13.14, 13.17 More exercises (3rd series)
    Exercises concerning planes, lines, systems of linear equations in previous years' exams
  • Week 7 (April 11 - 15)
    Topics: Conics: eccentricity and polar equation. Cartesian equation. Conics with a central symmetry.
    Reference: Apostol, Calculus Vol. I, Sections 13.20-15
    Assigned exercises: Apostol: Section 13.21, 13.24, exercises of Section 13.25 not involving Mathematical Analysis.
  • Week 8 (April 18 - 22)
    Topics: Vector-valued functions. Reparametrizations. Regular curves. Velocity vector and speed. Acceleration vector and normal vector. Variuos examples.
    Reference: Apostol, Calculus Vol. I, Sections 14.1-8
    Assigned exercises: Apostol: Section 14.4, 14.7, 14.9. Exercises of past exams on vector-valued functions.
  • Week 9 (April 25 - 29)
    Topics: rc=length. Curvature. Linear spaces: definition and examples. Linear subspaces.
    Reference: Apostol, Calculus Vol. I, Sections 14.10-12 (only the definition of arc-length and arc-length function), 14.14. Sections 15.1-6.
    Assigned exercises: Apostol: Section 14.13 and 14.15 Exercises of past exams on vector-valued functions involving arc-length and curvature. Section 15.5.
  • Week 10 (may 2-6)
    Topics: Linear combinations. Linear span. Linear independence and dependence. Finite-dimensional linear spaces. Bases and dimension of a finite-dimensional linear space. Examples of infinite-dimensional linear spaces. Inner products and norms.
    Reference: Apostol, Calculus Vol. I, Sections 15.7-9. !5.11.
    Assigned exercises: Apostol: Section 15.10 Exercises of past exams on vector-valued functions involving linear independence, bases and dimension.
  • Week 11 (may 9 - 13)
    Topics: Inner products: further examples. Orthogonal sets. Example: trigonometric polynomials. Orthogonal and orthonormal bases. Components with respect to orthogal bases. Gram-Schmidt orthogonalization. Orthogonal subspace. Orthogonal decomposition theorem and orthogonal projection onto a finite-dimensional linear subspace. Best approximation..
    Reference: Apostol, Calculus Vol. I, Sections 15.12, 15.14-16
    Assigned exercises: Apostol: Section 15.13 and 15.17. Exercises of past exams on inner products, orthogonal bases, orthogonalization and orthogonal decompostion.
  • Week 12 (may 16 - 20)
    Topics: Matrices. Matrix multiplication and its properties. A linear system as a matricial equation. Linear transformations: definition and examples. The linear transformation from V(n) to V(m) associated to a m x n matrix. Null-space and range. identification of the null-space and the range in the case of linear transformation from V(n) to V(m) associated to a matrix.

    Reference: Apostol, Calculus Vol. I, Sections 16.13,16.15, 16.1, 16.2
    Assigned exercises: Apostol: Sections 16.4, 16.16
  • Week 13 (may 23 - 29)
    Topics: Nullity + rank theorem. Rank of a matrix as the dimension of the subspace spanned by its rows and as the dimension of the subspace spanned by its columns. Rank, nullity and linear systems: Rouche'-capelli theorem. Computation of the rank of a matrix by means of gaussian elimination. One-to-one linear transformations and nullity. Invertible linear transformations and their inverses. Description of the preimages of elements of the target space of a linear trasformation. Linear transformations with preassigned values on the elements of a basis. Matrix representing a linear transformation with respect to a basis of the source space and a basis of the target space. Composition of linear transformations corresponds to matrix multiplication. Invertible matrices and their inverses. Theorem: a n x n matrix is invertible if and only if its rank is n.

    Reference: Apostol, Calculus Vol. I, Sections 16.3, 16.5, 16.7, 16.9, 16.10, 16.17, 16.18, 16.19
    Assigned exercises: Apostol: Section 16.8 and 16.12, 16.16 16.20. Exercises of past exams and tests on linear transformations and matrices.
  • Week 14 (only june 1)
    Topics: Determinants (summary). Definition as a multinear and alternating function with value equal to 1 on the identity matrix. Definition with Laplace expansions. Calculation of determinants via Gaussian elimination. Theorem: given a n x n matrix A, det(A) is non-zero if and only if rk(A)=n$. Product theorem for determinants. Determinant of the inverse matrix. Formula for the inverse matrix.
    Reference: Apostol, Calculus Vol. II, Chapter 3.
    Assigned exercises: Apostol, vol. II: Section 3.6, 3.11 and 3.17. Exercises of past exams on determinants.
  • Week 15 ( june 6-10)
    Topics: Eigenvalues, eigenvectors, eigenspaces. Eigenvalues as zeroes of the characteristic polynomial. Trace. Diagonalizationability. Diagonalization. Change of basis and similar matrices.
    Reference: Apostol, Calculus Vol. II, Chapter 4.
    Assigned exercises: Apostol, vol. II: Sections 4.8 and 4.10. Exercises of past exams on eigenvalues and eigenvectors.
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 2017.
Moreover, during the course, we plan to have six or sven 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 6/30 or 7/30 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.



Intermediate tests
  • Second test (april 7). text and solution of one version (there were three more different versions)



Exam results

  • First exam (june 27). text and solution (only version n.1) NEW! Corrected some typos and added an explanation.
  • Fourth session (sept 30). text and solution (miscalculation corrected in the solution of Exercise 1).
  • Extra session (nov 7). text and solution (miscalculation corrected in the solution of Exercise 1).


Previous two years:
 link 2013 -'14
link 2014 -'15

Text and solutions of previous years year written tests: