NMAI058 Linear Algebra 2

Summer Semestr 2025/26

Tuesday 15:40 - 17:10, Troja, Impact, Lecture Room N1

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 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).
• 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.
• Dates can be found in the SIS, registration there as well.

Tutorials:

• Volodymyr Kuznietsov
• Haochi Jiang

• February 17 Notes.
  • Introduction
  • Inner product
    • Motivation, definition
    • Norm, norm derived from inner product
    • Cauchy-Schwarz inequality
• February 24 Notes.
  • Orthogonality
    • Orthogonality. Linear independence of non-zero orthogonal vectors
    • Orthonormal basis, Fourier coefficients
    • Orthogonal projection
    • Gram-Schmidt orthonormalization
• March 3 Notes.
  • Orthogonality
    • Orthogonal complement
    • Properties of the orthogonal complement of a subspace
    • Least squares method
• March 10 Notes.
  • Orthogonality
    • Least squares method - missing proof
    • QR-decomposition of regular matrices as a consequence of Gram-Schmidt orthonormalization
  • Determinant
    • Motivation
    • Definition
    • Linearity
    • Swapping rows changes the sign
• March 17 Notes.
  • Determinant
    • Determinant of the transpose
    • Elementary row operations - determinants of their matrices; their effect on the determinant of the modified matrix
    • Computing determinant using Gaussian elimination
    • Determinant and matrix invertibility
    • det(AB) = det(A)det(B)
    • Cramer's rule
• March 24 Notes.
  • Determinant
    • Cramer's rule - proof
    • Inverse matrix formula
    • Laplace expansion
    • Adjugate matrix
  • Eigenvalues and eigenvectors
    • Motivating example
    • Similar matrices, diagonizable matrix
    • Eigenvalue of a linear transformation and associated eigenvectors
    • Eigenvalue of a matrix and associated eigenvectors
• March 31 Notes.
  • Eigenvalues and eigenvectors
    • Characteristic polynomial
    • A matrix is diagonalizable if and only if there exists a basis of eigenvectors
    • Similar matrices have the same characteristic polynomial
    • Important coefficients of the characteristic polynomial
• April 7 Notes.
  • Eigenvalues and eigenvectors
    • Every complex matrix is similar to an upper triangular matrix
    • Eigenvalues, determinant and diagonal entries of a matrix
    • Diagonalizability and algebraic and geometric multiplicity of eigenvalues
• April 14 Notes.
  • Eigenvalues and eigenvectors
    • Diagonalizability and algebraic and geometric multiplicity of eigenvalues - proof
    • Diagonalizability of symmetric matrices

Tentative Schedule

• April 21
  • Eigenvalues and eigenvectors
    • Graph diameter and eigenvalues of its adjacency matrix (without proof)
    • Jordan normal form
    • Characterization of bipartite graphs using eigenvalues of the adjacency matrix (one implication?)
  • Positive definite matrices
    • Definition, equivalent characterizations
    • Gram matrix
    • Sylvester's criterion
    • Recursive criterion
• April 28
  • Positive definite matrices
    • Testing using Gaussian elimination
    • Computing Cholesky decomposition using Gaussian elimination
    • Uniqueness of Cholesky decomposition
• May 5
  • Bilinear and quadratic forms
    • Diagonalization of symmetric bilinear forms
    • Sylvester's law of inertia for quadratic forms
• May 12
  • Bilinear and quadratic forms - signature
  • Linear algebra and GPT (I also recommend two nice videos by Grant Sanderson: GPT, Attention)
• May 19
  • Semester review - what we did and how it all connects

Various Sources

• Lecture Notes Linear Algebra 1 & 2 by Irena Penev, Charles University
• Slides and Videos Linear Algebra 1 & 2 by Jiri Fiala, Charles University
• Practice Assignments, Solutions here
• Lecture Notes Linear Algebra 1 by Bernd Gartner, ETH Zurich
• Lecture Notes Linear Algebra 2 by Afonso S. Bandeira and R. Weismantel, ETH Zurich
• Collection of Solved Problems, Charles University
• A series of videos Essence of Linear Algebra by Grant Sanderson - useful for getting an intuition for many key notions
• A series of videos MIT Linear Algebra by Gilbert Strang

April 14, 2026