COURSE AIMS AND OBJECTIVES: The first part of the course covers basic properties and techniques of unitary (inner product) spaces. The second, more extensive part, is devoted to linear operators, providing a deeper understanding of the concepts from Linear algebra 1, as well as various geometric and algebraic applications.
COURSE DESCRIPTION AND SYLLABUS:
I. UNITARY SPACES
I.1. Definition of unitary space (over fields R and C). Basic properties. Examples, in particular V2(O) and V3(O). Cauchy-Schwarz inequality. Gram matrix and determinant. Orthogonality relation for vectors. Orthogonal set. Subspace of vectors orthogonal to a given set.
I.2. Norm and normed space. Basic properties and examples of a norm. Orthonormal set. Norm induced by inner product. The parallelogram law. Metric (distance function) and metric space. Metric induced by norm.
I.3. Orthonormal basis of a unitary space. Expression of inner product, norm and metric in an orthonormal basis. Orthogonal projection on a given direction. Gram-Schmidt orthonormalization process. Orthogonal complement of a subspace of a finite-dimensional unitary space. Orthogonal projection on a subspace. Distance of a vector to a subspace. Least squares method.
II. LINEAR OPERATORS
II.1. Definition of linear operator (map). Basic properties. Examples, in particular on V2(O) and V3(O). Composition of linear operators and inverse of a bijective linear operator.
II.2. Defining a linear operator by its action on a basis of a finite-dimensional vector space. Matrix of a linear operator in a pair of bases. Matrix representation of the action on a vector. Reconstruction of a linear operator from its matrix representation. Matrix representation for some important examples of linear operators.
II.3. Action of linear operators with regard to subspaces. Kernel (null space) and image of a linear operator. Rank and deficiency (nullity). Characterization of injectivity by kernel. Monomorphism, epimorphism and isomorphism of vector spaces. Characterization by action on linearly independent sets, spanning sets and bases. The rank-deficiency theorem and its consequences. Interpretation of system of linear equations using linear operators. Isomorphism of vector spaces. Characterization of isomorphism of finite-dimensional vector spaces.
II.4. The space L(V,W) of linear operators. Algebra L(V). Matrix representation of composition of linear operators. Isomorphism of vector spaces L(V,W) and Mmn(F), resp. algebras L(V) and Mn(F). Group GL(V).
II.5. Linear functional. Dual space of a vector space. Dual basis. Description of linear functionals on a finite-dimensional unitary space by means of the inner product.
II.6. Matrix representation of a linear operator in various pairs of bases. Equality of rank of a linear operator and its associated matrix in any pair of bases. Similar matrices. Some invariants of similarity (rank, determinant, trace)
II.7. Eigenvalues and eigenvectors of a linear operator. Examples. Eigenspace. Spectrum of a linear operator. Characteristic polynomial and its zeroes. Algebraic and geometric multiplicity of an eigenvalue. Diagonalization of a linear operator. Necessary and sufficient conditions for diagonalization.
II.8. Operator polynomials. Hamilton-Cayley theorem. Invariant subspaces. Adjoint operator. (This part is concise and only informal).
II.9. Linear operators on unitary space. Unitary operator. Basic properties and examples. Spectrum of an unitary operator. Matrix of an unitary operator in orthonormal basis. Classification of unitary operators on unitary spaces V2(O) and V3(O). Symmetric and hermitian operators. Spectrum and diagonalization of symmetric and hermitian operators.
II.10. Some applications of linear operators: quadratic forms, curves and surfaces of 2nd order. Positive definite and semidefinite symmetric matrices. Extreme points of quadratic polynomials in n variables. Systems of recursive equations.