Introduction to the Theory of Computing 1

1) Coordinate geometry in the space: vectors in the space, coordinate system, scalar product. The equation of a plane. The (parametric and canonic) system of equations of a line. The notion of Rⁿ, operations on column vectors.

2) The notion of a subspace of Rⁿ, the property of being closed under operations. The notions of linear combination, generating system, spanned subspace. The notion of linear independence, the equivalence of the two definitions.

3) Relation between the sizes of linearly independent systems and generating systems in subspaces. The notions of basis and dimension, the unicity of the dimension. The unicity of representing a vector with respect to a basis.

4) Solving systems of linear equations with the Gaussian elimination. The notions of row reduction, echelon form and reduced echelon form. Relation between the number of equations and the number of variables of uniquely solvable systems.

5) The definition of the determinant. The number of inversions in permutations. Basic properties of the determinant. Computing the determinant with the Gaussian elimination. Characterizing the unique solvability of (n x n) systems of linear equations with the determinant. Determinant expansion by minors.

6) The cross product and the mixed product of 3-dimensional vectors. The relation of the mixed product to the determinant. Operations on matrices, identity matrix, transposed matrix. The determinant of the product matrix. Expressing systems of linear equations on the Ax=b form. Connections between the linear independence of the rows and the columns

7) The notion of the inverse matrix, necessary and sufficient condition on its existence. Computing the inverse. The notion of the rank of a matrix, equality of the three types of rank notions. Computing the rank of a matrix.

8) The notion of a linear map, necessary and sufficient condition on the linearity of a map. The composition of linear maps, addition formulas for the sin and cos functions. The kernel and the image of linear maps, the rank-nullity theorem.

9) Basis transformation, the matrix of a linear transformation with respect to a basis, computing this matrix. The notions of the eigenvalue and the eigenvector of a square matrix, computing the eigenvalues, characteristic polynomial.

10) The basic notions of number theory: divisibility, prime numbers, the fundamental theorem of arithmetics, the cardinality of primes, the gap between adjacent primes, the prime number theorem. The notion of congruence, operations on congruences. Solvability of linear congruence equations.

11) Euclidean algorithm for computing the greatest common divisor and for solving linear congruence equations. Linear diophantine equations on two variables, simultaneous linear congruences. Euler's totient theorem, Fermat's little theorem.

12) Arithmetic algorithms: relation between the size of the input and the logartihm of the input data, basic operations, exponentiation modulo m, primality testing. Public key cryptography, the RSA code.

13)  Cardinality of infinite sets: the notions of equal and less than or equal cardinalities. The notions of sets of countably infinite and continuum cardinalities. The cardinality of the sets N, Z, Q and R.