Exam topics for Intro to CS I

January 2008

  1. Geometry: description of plane and line by linear equations; equations of parallel planes, intersections.
  2. Vector space, subspace, linear combination, generating system, subspace spanned by vectors.
  3. Linear independence, basis, dimension.
  4. Gauss elimination. Sovability of a system of linear equations. Condition to have unique solution.
  5. Determinant: definition, basic properties, its computation
  6. Matrix, operations with matrices. Inverse of a matrix. Rank.
  7. Connection between systems of linear equations and matrices, solvability and matrix rank, unique solution and determinant.
  8. Linear map (definition, basic properties). Matrix of a linear map.
  9. Kernel and image of linear maps. Dimension theorem.
  10. Eigenvalues and eigenvectors.
  11. Complex numbers, operations on complex numbers.
  12. Combinatorics: permutations, variations, combination (without and with repetitions); Binomial coefficients and their properties, Binomial theorem.
  13. Graphs, subgraphs, connected graphs; isomorphism.
  14. Trees, properties of trees; Prufer code, Cayley theorem
  15. Planar graphs, connection with drawing on the sphere; Euler theorem, Kuratowski theorem.