66876 - Linear Algebra

Learning outcomes

At the end of the module, the student understands and is able to use the basics of linear algebra: he is able to solve linear systems, solve problems in vector spaces and with the aid of vector spaces (in particular knows the concepts of linear independence, subspace, basis and dimension), multiply and invert matrices, study linear mappings (in particular knows the concepts of eigenvalue and eigenvector).

Course contents

• Systems of linear equations and their matrices; Gauss algorithm.
• Vector spaces of n-tuples R^n, geometric interpretation for n=1,2,3. Vector spaces on R; linear independence, bases and dimension; vectors and n-tuples. Dot product on R^n; geometric interpretation for n=1,2,3. Euclidean vector spaces; orthonormal bases. Subspaces; subspaces associated to a matrix and their dimensions; rank of a matrix.
• Matrices as mappings from R^n to R^m; geometric interpretation for n,m=1,2,3; algebra of matrices; inverse matrix; determinant. Linear mappings between vector spaces; algebra of linear mappings. Linear mappings and matrices. Orthogonal matrices. Isomoetries between euclidean vector spaces. Subspaces associated to a linear mapping.
• Endomorphisms of a vector space; eigenvectors and eigenvalues; diagonalizability and diagonalization. Symmetric matrices. Self-adjoint endomorphisms of an eucliean vecor space; spectral theorem.

Readings/Bibliography

• Online lecture notes and exercises, weekly published during the course by the lecturer on Virtuale.
  

Teaching methods

• Lectures.
• Every week exercises will be given, that will be corrected by a tutor.
  
  

Assessment methods

In order to pass the exam of the integrated course Mathematical Analysis - Linear Algebra on must pass the exam on each of the two parts; the grade of the integrated course is the mean of the grades of the two parts.

Linear Algebra exam:

• Written and oral exams, to be taken in the same "appello". In order to attend the oral exam one must achieve at least a grade 12/30 in the written exam. The oral exam is crucial.
  
• The written exam aims at verifying the ability of solving exercises akin to those assigned during the course. The steps leading to the solution must be given and justified. It is not allowed to use books, notes or calculators; only paper and pen. It lasts 1 hour and 30 minutes.
• The oral exam aims at verifying the knowledge of the theory developed during the course. It will be asked to give definitions and examples of concepts and to give statements and proofs of propositions. Il lasts about half an our.

Teaching tools

Further material will be published during the course on Virtuale.

Office hours

See the website of Francesco Regonati