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 acquainted with Euclidean vectors, numerical vector spaces Rn (n=1,2,3,...), abstract vector spaces and is able to compute in Rn, interpret, discuss and solve linear systems. He is acquainted with geometric transformations, matrix algebra, linear mappings and is able to compute with matrices and to use eigenvalues and eigenvectors.

Course contents

1. Finite dimensional vector spaces. Geometric vectors and operations on them. Real vector spaces. Geometric vector spaces Vn (n=1,2,3), numeric vector spaces Rn (n=1,2,3,4,...), function vector spaces (allusion). Linear combinations. Linear independence. Vector spaces with a finite basis, coordinates, identification with a space Rn; dimension. Geometric vectors, length, orthogonality, scalar product. Euclidean vector spaces. Euclidean vector spaces En (n=1,2,3), Rn (n=1,2,3,4,...), function Euclidean spaces (allusion). Orthogonal projections, orthogonal bases.
2. Matrix algebra. Row space and column space of a matrix; Gauss and Gauss-Jordan algorithms; bases of the row space and column space; rank. Sum of matrices; product of rows by columns; product of matrices; properties. Algebra of square matrices; determinant; invertibility and inversion. Orthogonal matrices.
3. Linear systems. Linear systems of n equations in n unknowns with a unique solution. Elimination method. Cramer rule. Homogeneous linear systems, null space of a matrix, a basis, dimension. Solvability and structure of the solution set. Linear systems as instances of matrix equations.
4. Linear mappings between finite dimensional vector spaces. Linear mappings between vector spaces Rn and matrices; structure of a linear mapping and linear systems; composition, inversion of linear mappings and product, inversion of matrices. Linear mappings between vector spaces; matrix with respect to bases of codomain and domain; relation between the matrices of one endomorphism.
5. Eigenvectors, eigenvalues, diagonalization. Geometric vector transformations, invariant vector straight lines. Endomorphisms of a vector space, eigenvectors, eigenvalues, diagonalizability. Characteristic polynomial, eigenspaces of an endomorphism. Theorems on diagonalization. Spectral theorem on orthogonally diagonalizable endomorphisms of an Euclidean vector space.
   
   

Readings/Bibliography

• Online lecture notes and exercises, weekly published during the course by the lecturer on Virtuale.
• For an opening on a landscape of contents and applications wider that that of the course, see: G. Strang, Linear Algebra for Everyone, Wellesley-Cambridge Press.

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 one 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