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- Sets. equality of sets. subsets. operations on sets. functions, one-to-one functions, onto functions, permutations. composition of functions, inverse function.
- Algebraic systems: Algebraic operations on a set, properties of algebraic operations – commutativity, associativity, identity element, inverse element.
- Groups: definition of a group, commutative and non-commutative groups. Various examples of groups: groups of numbers (integers, rationals, reals), group of permutations of a finite set, group of bijections of any set, groups of residues modulo n with addition and with multiplication – in the case of prime n. Subgroups – definition, examples. The concept of isomorphism of groups and algebras in general.
- Fields: definition of a field, arithmetic properties of operations in a field. Examples of fields – the field of rational numbers, real numbers and complex numbers, finite fields of residues mod n. Isomorphism of fields. Examples of isomorphic and of non-isomorphic pairs of fields. Conjugation as a nontrivial automorphism of the field of complex numbers.
- Polynomials over a field. The long division algorithm. The remainder theorem. Roots of a polynomial. Main Theorem of Algebra and its consequences for polynomials with real coefficients.
- Vector spaces: definition of a vector space over a field. Examples of vector spaces. The concept of a subspace. The subspace spanned by a set of vectors. Linear combination of a finite set of vectors. Linear independence of a finite set of vectors. The replacement lemma. Basis and dimension of a vector space. Finite dimensional vector spaces.
- Matrices. Definition of a matrix as a function. Matrix addition, scaling and multiplication. Transposition of a matrix. Square matrices. Symmetric matrices. Rows and columns of a matrix as vectors. Rank of a matrix as the dimension of the space spanned by rows. Elementary row operations. Row echelon and row canonical forms of a matrix.
- Systems of linear equations. General and homogeneous systems. Matrix form of a system of linear equations. System of linear equations as a linear combination of columns of the coefficient matrix. The solution space for a homogeneous system. Kronecker-Capelli theorem. The Gauss reduction method.
- Determinant of a matrix. Laplace expansion. Determinants versus linear independence. Using determinants in solving systems of linear equations – the Cramer’s Rule. Nonsingular matrices. The inverse matrix.
- Linear mappings: definition and examples. The matrix representation of a linear mapping with respect to given bases. Linear operators. Similar matrices. The change-of-basis matrix.
- Eigenvalues and eigenvectors of a linear operator. Characteristic polynomial of a matrix. Invariant subspaces. The notion of the direct sum of subspaces of a vector space. Jordan canonical form of a matrix.
- Lines and planes in the 3-dimensional space.