1. komponenta

Lecture type	Total
Lectures	60
Exercises	45

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

COURSE AIMS AND OBJECTIVES:
The aim of the course is to familiarize students with the elements of the theory of linear operators on finite-dimensional vector (inner product) spaces.

COURSE DESCRIPTION AND SYLLABUS:
Linear operators. Definition and basic properties of linear operators. Examples of linear operators on R2 and R3 . Defining a linear operator by its action on a basis. The space of linear operators L(V,W). Basis and dimension of L(V,W). Dual space. Dual basis. Linear functionals on Rn . Kernel and image of a linear operator. Rank and nullity of an operator. Monomorphisms, epimorphisms, and isomorphisms. Isomorphic spaces. The rank-nullity theorem. Matrix representation of vectors and linear operators. Reconstruction of an operator from its matrix representation. Matrix representation of the composition of linear operators and inverses. Matrix representations in different bases. Change-of-basis matrix. Rank of an operator and the corresponding matrix. Examples.
Eigenvalues and eigenvectors. Definition of eigenvalues and eigenvectors. Examples (rotation and reflection with respect to a line in the plane, orthogonal projection, differentiation operator). The concept of diagonalizability of a linear operator. Characteristic polynomial of a matrix and a linear operator. Spectrum of a linear operator. Upper triangular matrix representation of a linear operator (Schur theorem). Diagonalization of a linear operator. Eigenspaces. Geometric and algebraic multiplicity of an eigenvalue. Characterization of diagonalizability of a linear operator over complex and real vector spaces. Eigenvalues and eigenvectors of a matrix. Matrix diagonalization. Cayley-Hamilton theorem. Examples. Application in internet search. Complex zeros of the characteristic polynomial of a real matrix.
Inner product spaces. Examples. Cauchy-Schwarz inequality. Norm. Orthonormal basis. Gram-Schmidt orthogonalization. Orthogonal complement. Orthogonal projector. QR factorization.
Linear operators on inner product spaces. Linear functionals on inner product spaces. Riesz representation theorem for linear functionals. Hermitian adjoint operator. Matrices of operators A and A* in an orthonormal basis. Approximation problems. Least squares solutions. Unitary operators. Characterizations of unitary operators and examples. Unitary operators on R2 . Diagonalization of a linear operator in an orthonormal basis. Hermitian operators and their properties. Normal operators and their properties. Spectral theorem on complex inner product spaces. Spectral theorem on real inner product spaces. Polar form and singular value decomposition. Conic sections and quadratic forms.

1. Linearna algebra, Lj. Arambašić, Linearna algebra, 2022.
2. Linear algebra done right, S. Axler, Springer-Verlag, New York, 1997.
3. Linearna algebra, D. Bakić, Školska knjiga, 2008.
4. Linear algebra and its applications, D. Lay, S. Lay, J. McDonald, Pearson, 2016.
5. Linear algebra, a modern introduction, D. Poole, Brooks/Cole, 2011.
6. Linear algebra and its applications, G. Strang, Saunders College Publ, 1986.

Enrollment :
Attended : Linear algebra 1

Examination :
Passed : Linear algebra 1

Mandatory course - Regular study - Mathematics