NMAI057 Linear Algebra 1

Winter Semestr 2025/26

Thursday 10:40 - 12:10, Malá Strana, Lecture Room S9

Office Hours during the teaching period in semester: Tuesday 11:00 - 12:00, Malá Strana, office 225, or by appointment.

• The course credit is a prerequisite for taking the exam.
• The exam typically has two parts: written and oral. The written part includes examples, definitions, formulation of statements, and proofs (basic mathematical skills are assumed – working with propositions and quantifiers, sets, various proof techniques).
• Students who successfully complete the written part proceed to the oral part. It takes the form of a discussion on the assigned topic. This is an iterative process where the examiner typically asks supplementary questions.
• Dates can be found in the SIS, registration there as well.

Tutorials:

• Todor Antić
• Adam Morawski
• Sasha Sami

• October 2 Notes.
  • Introduction.
  • Systems of linear equations.
  • Elementary row operations.
• October 9 Notes.
  • Elementary row operations do not change the solution set.
  • Row echelon form of a matrix.
  • Row reduction algorithm.
  • Solving a system of linear equations.
• October 16 Notes.
  • Uniqueness of pivot positions.
  • Rank of a matrix.
  • Reduced row echelon form of a matrix.
  • Gauss-Jordan elimination.
  • Relationship between solution sets of homogeneous and non-homogeneous systems.
• October 23 Notes.
  • Properties of matrix addition and scalar multiplication.
  • Binary operations and groups.
  • Matrix multiplication and its properties; transpose; identity matrix.
  • Inverse of a matrix, invertible matrix, and connection to matrix rank.
• October 30 Notes.
  • Proof of associativity of matrix multiplication.
  • Proof of the connection between matrix invertibility and rank.
  • Matrices of elementary row operations, and of sequences of EROs.
  • Permutations. Permutation group.
• November 6 Notes.
  • Permutations. Inversion pairs. Sign of a permutation. Transpositions.
  • Fields.
  • Finite fields Z_p.

Tentative Schedule

• November 13
  • Fermat’s little theorem. Characteristic of a field.
  • Vector spaces. Subspaces. Linear combinations of vectors.
  • Linear span. Linear dependence and independence.
• November 20
  • Spaces determined by a matrix (row space, column space, kernel(A), kernel(A^T)).
  • Generating set. Basis.
  • Multiplication by a regular matrix on the left does not change the row space and the kernel.
  • Exchange lemma and Steinitz exchange theorem.
• November 27
  • Dimension of a vector space.
  • Dimensions of spaces determined by a matrix and rank of a matrix.
  • Multiplication by a regular matrix on the left does not change the dimension of the column space.
  • Transposition does not change the rank of a matrix.
• December 4
  • Coordinates of a vector relative to a basis.
  • Union of two subspaces. Dimension of union and intersection.
  • Linear mappings. Basic properties. Examples.
  • Isomorphism. The theorem “every vector space of dimension n over F is isomorphic to F^n”.
• December 11
  • Images of a basis uniquely determine a linear mapping.
  • Matrix of a linear mapping relative to bases.
• December 18
  • Composition of linear mappings, matrix of the composed mapping.
  • Isomorphism and its matrix.
• January 8, 2026
  • Affine subspaces, connection with solutions of non-homogeneous systems of linear equations.
  • Additional topic: change of basis and compression of the image – outline of applications.
  • Review

November 6, 2025