Academic year 2020-21
Timetable of the lectures (Prof. Salvatore and Dr. Lhotka): Mon 4pm, Thu 9.30am, Fr 9.30am
Timetable of tutorials (Dr. Sabatino): Fri 3pm
Lectures online on Microsoft Teams, channel "Lectures 2021", team "Linear Algebra and Geometry". Code of the Team: nsga0pq
Exams: written and oral examination.
Written exams dates: June 21 and July 8, September 1 and 15, 2-4pm
Guidelines for the written exam:
1) Register on Delphi for the relevant exam
2) Join the Linear Algebra and Geometry team in Microsoft Teams using the code nsga0pq
3) On the day of the exam join the meeting on the channel "Written exams" of the team. Turn on your webcam and microphone so to be visible. You must leave them on for the whole duration of the exam.
4) Click on the moodle dashboard resource in the channel "Written exam". This will connect you to the course dashboard
5) Log in using Open ID connect, NOT username and password !
6) Click on the exam of the day and follow the instructions
7) When you are finished, submit your answers and leave the meeting.
8) In case of any problems use the private chat to report them.
All notes of the lectures are online in the channel "Lectures 2021", team "Linear algebra and geometry". The lectures are recorded and available.
Suggested books: Lay, Linear algebra and its applications; Bretscher, Linear algebra with applications; Apostol, Calculus vol. 1,2
Lectures 2019
4-3-19: Linear equations and linear systems. Solutions. Consistency of a system.
7-3-19: Basic and free variables. Matrix of coefficients. Augmented matrix. Row reduction to echelon matrix.
8-3-19: Exercises on linear systems.
11-3-19: Numerical vectors. Addition and multiplication by scalars. Linear combinations. Linear systems and vectors.
15-3-19: Linearly independent vectors. Finding subsets of linearly independent vectors.
16-3-19: Linear systems in matrix form. Exercises on linear systems in vector form.
18-3-19: Canonical basis. Linear space. Basis and coordinates of vectors.
21-3-19: Steinitz lemma. Dimension of linear spaces. Rank of a matrix.
22-3-19: Linear spaces of rows and columns of a matrix. Null space of a matrix.
25-3-19: Matrix transformations. Injectivity, surjectivity and rank.
28-3-19: Linear transformations and matrices. Examples.
29-3-19: Multiplication and addition of matrices and their linear transformations.
1-4-19: Invertible matrices.
4-4-19: Computing the inverse via row reduction
5-4-19: Change of coordinates and matrices
8-4-19: Vector (linear) spaces. Examples of polynomials and matrices.
11-4-19: Linear subspaces. Intersection of linear subspaces.
12-4-19: Exercises
15-4-19: Sum of linear subspaces. Grassmann formula.
18-4-19: Basis for intersections and sums of linear spaces.
29-4-19: Midterm exam
2-5-19: Determinants: definition, properties, computation.
3-5-19: Computation of the rank using determinants.
Computation of the inverse matrix using determinants. Determinant of a product.
Cramer's formula.
6-5-19: Linear transformation between vector spaces. Image and kernel.
9-5-19: Matrix of a linear transformation with respect to basis of the domain and of the range.
10-5-19: Lines in the plane and in 3-dim. space.
Planes in the 3-dim. space. Cartesian and parametric equations. Lines through 2 points. Plane through 3 non collinear points. Relative position of two planes.
13-5-19: Relative position of two lines in 3-dimensional space.
16-5-19: Inner product. Norm. Distances. Orthogonal vectors, lines, planes. Angles.
17-5-19: Cross product in 3-dim. space. Mixed product. Area of parallelogram. Volume of parallelepiped.
20-5-19: Eigenvalues and eigenvectors. Characteristic polynomial.
23-5-19: Algebraic and geometric multiplicities
24-5-19: Diagonalization of endomorphisms and matrices
30-5-19: Orthogonal subspaces, orthonormal basis, orthogonal matrices.
31-5-19: Gram-Schmidt orthonormalization. Formula for the orthogonal projection.
3-6-19: Matrix of orthogonal projections. Spectral theorem for symmetric matrices.
6-6-19: Quadratic forms and their classification
7-6-19: Conic curves: classification
10-6-19: Rotations and translations that put a conic in normal form
13-6-19: Exercises
14-6-19: Exercises