Exam topics for Intro to CS I
January 2008
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Geometry: description of plane and line by linear equations;
equations of parallel planes, intersections.
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Vector space, subspace, linear combination, generating system,
subspace spanned by vectors.
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Linear independence, basis, dimension.
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Gauss elimination. Sovability of a system of linear equations.
Condition to have unique solution.
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Determinant: definition, basic properties, its computation
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Matrix, operations with matrices. Inverse of a matrix. Rank.
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Connection between systems of linear equations and matrices,
solvability and matrix rank, unique solution and determinant.
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Linear map (definition, basic properties). Matrix of a linear map.
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Kernel and image of linear maps. Dimension theorem.
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Eigenvalues and eigenvectors.
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Complex numbers, operations on complex numbers.
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Combinatorics: permutations, variations, combination (without and with
repetitions); Binomial coefficients and their properties, Binomial theorem.
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Graphs, subgraphs, connected graphs; isomorphism.
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Trees, properties of trees; Prufer code, Cayley theorem
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Planar graphs, connection with drawing on the sphere;
Euler theorem, Kuratowski theorem.