Transition Math and Algebra with Geometry (ETMAG 25z)🔗

Announcements🔗

  • 2025-10-01: Announcements related to this course will appear here.

Timetable🔗

Activity  Group     Time            Room        Lecturer

Lecture   all       Mon.  8:15-10   162         Gabriel Pietrzkowski, PhD
Tutorial  101       Thu. 14:15-16   9A          Tomasz Kostrzewa, PhD
Tutorial  102       Thu. 10:15-12   104         Tomasz Kostrzewa, PhD

Content🔗

  1. Sets and functions: basic definitions and properties. "One-to-one" functions, "onto" functions and permutations. Function composition and inversion.

  2. Functions and graphs. Exponential and logarithmic functions. Inverse trigonometric functions.

  3. Complex numbers: definition and properties. Polynomials over the field of real and complex numbers. Remainder theorem. Roots of polynomials. Main Theorem of Algebra.

  4. Matrices: basic definitions and properties. Matrix operations: addition, scaling, multiplication, transposition. Square matrices, symmetric matrices.

  5. Determinant of a matrix: definition and properties. Rank of a matrix.

  6. System of linear equations: General and homogeneous systems. Matrix form of a system of linear equations. Cramer's rule. Kronecker-Capelli theorem. Gauss elimination.

  7. Nonsingular matrices: matrix inversion.

  8. Vector spaces: basic definitions. Examples of vector spaces over the field of real and complex numbers. Subspaces: definitions and properties. Linear independence of a finite set of vectors in finite dimensional vector spaces. Basis and dimension of a vector space: definition and properties. Solution space of a system of linear equations.

  9. Axioms for real numbers. Sequences. Arithmetic and geometric sequences. Monotonicity and boundedness of a sequence. Convergence and divergence. Properties of the limit.

  10. Basic convergent sequences. The number e. Evaluating limits.

  11. Limits: one-sided limits and limits at infinity; continuity, theorems on continuous functions; extreme values.

  12. Derivatives: definitions, physical and geometric interpretation. Rolle's theorem. Lagrange's theorem. Cauchy's theorem.

  13. Evaluation of derivatives. Applications of derivatives. De l'Hospital rule.

  14. Derivatives of higher order: graphing functions using y' and y''. Monotonicity and convexity.

Textbooks🔗

  1. S. Lipschutz, M. Lipson: Schaum`s Outline of Linear Algebra, McGraw-Hill; 6rd edition (2017)

  2. Joel R. Hass, Christopher E. Heil, Maurice D. Weir: Thomas' calculus, Pearson; 14th edition (2018).

  3. ED. Bloch, The real numbers and real analysis. Springer Science & Business Media (2011)

Course regulations🔗

  1. Academic Regulations at the Warsaw University of Technology.

  2. Course regulation in USOSweb an in the document.

Lecture notes🔗

Lecture notes Theory Solutions Tutorial class
Symbols Suite 1
Suite 2 NEW
Basic set theory Lecture 1 Theory 1 Solutions 1
Functions Lecture 2 Theory 2 Solutions 2 Suite 3 NEW
Suite 4 NEW
Complex numbers Lecture 3 Theory 3 Solutions 3 Suite 5 NEW
Polynomials Lecture 4 Theory 4 Solutions 4 Suite 6 NEW
Midterm test
Matrices Lecture 5 Theory 5 Solutions 5 Suite 7 NEW
Systems of Linear Equations Lecture 6 Theory 6 Solutions 6 Suite 8 NEW
Matrices: Invertability Lecture 7 Theory 7 Solutions 7
Vector spaces Lecture 8 Theory 8 Solutions 8
Sequences Lecture 9 Theory 9 Solutions 9 Suite 9 NEW
Limits of functions, Asymptotes Lecture 10 Theory 10 Solutions 10 Suite 10 NEW
Suite 11 NEW
Continuous functions Lecture 11 Theory 11 Solutions 11 Midterm test
Differentiable functions Lecture 12 Theory 12 Solutions 12 Suite 12 NEW
Applications of differtiability Lecture 13 Theory 13 Solutions 13 Suite 13 NEW
Convexity and Graph Lecture 14 Theory 14 Solutions 14