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

 

1. Finite-dimensional vector spaces.
   
   Geometric vectors; frames. Real vector spaces. Geometric vector spaces Vn (n=1,2,3). Numeric vector spaces Rn (n=1,2,3,4,...).
   
   Linear combinations; linear independence; dimension and bases of a vector space; coordinates with respect to a basis.
   
   Length, orthogonality, dot product of geometric vectors. Euclidean vector spaces. Euclidean vector spaces Rn. Orthogonal bases.
   
   
2. Linear systems and matrices.
   
   Linear systems and matrices. Gaussian elimination. Linear systems of n equations in n unknowns with a unique solution and bases of Rn.
   
   Matrices, product, invertibility, inversion. Description Ax=b of a linear system. Invertibile matrices and linear systems with a unique solution.
   
   Signed areas and volumes. Determinant of an nxn matrix. Determinant and invertiblity of matrices.
   
   
3. Subspaces
   
   Vector lines and vector planes in V3. Subspaces of a vector space. Spaces associated to a matrix; rank, dimension theorem. Structure of solutions of a linear system.
   
   
4. Linear mappings between finite-dimensional vector spaces.
   
   Trasformations of geometric vector spaces. Linear mappings between vector spaces. Linear mappings between numeric vector spaces Rn and matrices.
   
   Endomorphism of a vector space, matrix with respect to a basis; composition of endomorphisms and product of matrices. Relation between matrices of the same endmorphism.
   
   Linear mappings between vector spaces: injectivity, bijectivity, invertibility; kernel space, image space, dimension theorem.
   
   
5. Eigenvectors, eigenvalues, diagonalization.
   
   Geometric transformations, invariant vector lines. Eigenvectors, eigenvalues, diagonalizability of an endomorphism.
   
   Characteristic polynomial, eigenspaces. Theorems about 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 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