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Number theory1P (mm1c2se1)
Syllabus designed by:
Classical algebra (complex numbers, polynomials, matrix operations, determinant).
- P. M. Cohn: Classic algebra. Wiley, 2000
- Vector spaces: axioms, examples. Generated subspace, linear dependence and independence, basis, dimension, coordinates.
- The Exchange Theorem, the dimension is well-defined. The dimension of a proper subspace. The sum and direct sum of subspaces.
- Linear mappings and transformations: homomorphisms between vector spaces. Addition, multiplication by a scalar, composition. The concept of algebra. The matrix of a linear mapping in a given pair of bases, relationship to matrix operations. Formula to compute the matrix in a new pair of bases.
- Kernel and range, the connection between the dimensions of these subspaces. Invertible transformations, zero divisors. The dimension of the space of linear mappings. Isomorphic vector spaces and algebras. The determinant of a linear transformation.
- The rank of a system of vectors and of linear mappings. The characterizations of the rank of a matrix using columns, rows, and nonzero minors. Computing the rank using Gaussian elimination. The rank and systems of linear equations.
- The diagonal form of a linear transformation and of a square matrix. Eigenvalues, eigenvectors, eigenspaces, the characteristic polynomial.
- The minimal polynomial of a transformation. The Cayley-Hamilton Theorem. The zeroes of the minimal polynomial are exactly the eigenvalues.
- Invariant subspaces. A transformation has a diagonal form over a field if and only if its minimal polynomial is a product of linear factors over this field, and has no multiple roots. Jordan normal form, uniqueness (without proof). The powers of Jordan-blocks.
- Euclidean spaces over the real and complex numbers, orthonormal basis, orthogonal complement of a subspace. Angle and norm of vectors, the Cauchy-Schwarz inequality, the triangle inequality.
- The adjoint of a linear transformation, its matrix. Normal, self-adjoint and symmetric, unitary and orthogonal transformations, their eigenvalues. Theorem of Principal Axes.
- Bilinear, symmetric bilinear and Hermitian bilinear functions, their matrix. Change of basis. Orthogonality, the Gram-Schmidt method. The character of a quadratic form, test using the principal minors. Sylvester's law of inertia.