1. komponenta

Lecture type	Total
Lectures	45
Exercises	30

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES:
Introduction to the concepts of inner product spaces and linear operators. Generalization of well-known geometric examples and results to general inner product spaces. Deepening and connecting topics previously studied in Linear Algebra 1 (vector spaces, matrices, systems of linear equations) through the theory of linear operators.

COURSE DESCRIPTION AND SYLLABUS:
1. Definition of an inner product space (over R and C). Basic properties. Examples, especially V2(O) and V3(O). Cauchy-Schwarz inequality.
2. Gram matrix and Gram determinant. Orthogonality relation. Orthogonal sets. Subspace of vectors orthogonal to a subset.
3. Norm and normed spaces. Basic properties and examples. Orthonormal sets. Norm induced by an inner product. Parallelogram law. Metric and metric spaces. Metric induced by a norm.
4. Orthonormal bases. Expression of inner products, norms and metrics in an orthonormal basis. Orthogonal projection onto a line. Gram-Schmidt orthogonalization. Orthogonal complements.
5. Orthogonal projection onto a subspace. Distance from a vector to a subspace. Least-squares method.
6. Definition and properties of linear operators. Examples. Composition and inverses.
7. Matrix representation of linear operators. Reconstruction from matrix representations.
8. Kernel, image, rank and nullity. Injectivity, surjectivity and isomorphisms. Rank-nullity theorem and applications.
9. Spaces and algebras of linear operators. Relation with matrix spaces and matrix algebras.
10. Linear functionals. Dual spaces and dual bases.
11. Matrix representations in different bases. Similar matrices and similarity invariants.
12. Eigenvalues, eigenvectors, eigenspaces, spectrum, characteristic polynomial and diagonalization.
13. Operator polynomials. Cayley-Hamilton theorem. Invariant subspaces. Adjoint operators.
14. Unitary operators. Symmetric and Hermitian operators. Spectra and diagonalization.
15. Applications: quadratic forms, conics and quadrics, positive (semi)definite matrices, extrema of quadratic polynomials, recursive systems.

1. Linearna algebra, Z. Franušić, J. Šiftar, PMF, Zagreb, 2022.
2. Linearna algebra, Lj. Arambašić, Element, Zagreb, 2022.
3. Linearna algebra i primjene, D. Bakić, Školska knjiga, Zagreb, 2021.

Enrollment :
Passed : Linear algebra 1

Mandatory course - Regular study - Mathematics and Physics Education