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.
Exam - information
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The course credit is a prerequisite for taking the exam.
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The list of topics you need to know for the exam is given here.
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The exam typically has two parts: written and oral.
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The written part takes 90 minuts. It includes problems, definitions and examples, formulation of statements, and proofs (basic mathematical skills are assumed – working with propositions and quantifiers, sets, various proof techniques). Sample exam.
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Students who successfully complete the written part proceed to the oral part (typically in the afternoon).
It takes the form of a discussion on the assigned topic. This is an iterative process where the examiner typically asks supplementary questions.
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Dates can be found in the SIS, registration there as well.
Tutorials:
Covered Topics
- 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.
- November 13 Notes.
- Fermat’s little theorem. Characteristic of a field.
- Vector spaces.
- Linear combinations of vectors.
- November 20 Notes.
- Vector spaces - basic properties.
- Subspaces. Linear span of a subset.
- Intersection of subspaces.
- Linear span as an intersection of subspaces.
- November 27 Notes.
- Spaces associated with a matrix (row space, column space, nul(A), nul(AT)).
- Multiplication by a regular matrix from the left does not change the row space and the null space.
- Linear dependence and independence.
- Spanning collection. Basis.
- Exchange lemma.
- December 4 Notes.
- Steinitz exchange theorem.
- Dimension of a vector space.
- Coordinates of a vector relative to a basis.
- Dimensions of spaces associated with a matrix and the rank of a matrix.
- Multiplication by a regular matrix from the left does not change the dimension of the column space.
- December 11 Notes.
- Transposition does not change the rank of a matrix.
- Sum of two subspaces. Dimension of sum and intersection.
- Linear transformations (mappings). Examples. Basic properties.
- December 18 Notes.
- Isomorphism. The theorem “every vector space of dimension n over F is isomorphic to F^n”.
- Images of a basis uniquely determine a linear transformation.
- Matrix of a linear transformation with respect to bases.
- Change of basis matrix (transition matrix).
- Composition of linear transformations.
- January 8, 2026 Notes.
- Matrix of the composed transformation.
- Isomorphism and its matrix.
- 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
Various Sources
January 8, 2026