Linear algebra

Introduction to the finite dimensional vector spaces, linear operators, spectral theory of linear operators, analitical geometry.

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1. Matrices
   (addition and multiplication of matrices; reduction of matrices; rank of a matrix; triangular matrices; determinants and their properties; the Laplace expansion; inverse matrices; the general linear group GL(n))

2. Systems of linear equations
   (Gauss elimination method; equivalent linear systems; the Rouche'-Capelli theorem; homogeneous systems.)

3. Vector spaces
   (definition and basic properties; subspaces; linear combinations; linear independence; the direct sum of subspaces; basis and dimension; Grassmann theorem.)

4. Euclidean vector spaces
   (definition of scalar product and basic properties; bilinear forms; the isotropic cone; orthonormal basis; Gram-Schmidt method; orthogonal projections; orthogonal matrices and isometries.)

5. Linear maps (transformations) (definition and basic properties; kernel and image; isomorphisms; matrix of a linear transformation; composition of linear maps; change of basis in a vector space.)

6. Eigenvalues and eigenvectors
   (definition and characteristic polynomial; similar matrices; diagonalization of an endomorphism; the Jordan normal form; diagonalization of symmetric matrices.)

7. Notions of Affine Linear Geometry
   (Lines and planes in the affine space; Euclidean affine spaces; cross and mixed product; orthogonality and distance between linear affine varieties.)

Students will be able to:
- do operations with matrices
- calculate determinants
- solve linear systems
- determine a basis and the dimension of a vector space
- determine the coordinates of a vector with respect to a basis of a vector space
- calculate the kernel and image of a linear transformation
- find the eigenvalues and eigenvectors of a matrix
- diagonalize a symmetric matrix
- solve problems with vectors, lines and planes in the space

• P. R. Halmos, "Finite-Dimensional vector spaces" (Springer, 1993). Catalogue E-version
• G. Landi and A. Zampini, "Linear algebra and analytic geometry for physical sciences" (Springer, 2018). Catalogue

• written tests, written exam
• oral exam

Dr. Irina Elena Cristea je izredna profesorica za področje matematike na Univerzi v Novi Gorici.
Dr. Irina Elena Cristea is an Assistant professor of mathematics at the University of Nova Gorica.