Summary

1. General background: sets and operations on them.Complex numbers: definition (via ordered pairs), addition and multiplication, inverses, adjoint, absolute value, Real and complex polynomials and their roots.2. FieldsDefinition, properties, examples: Rationals, reals, complex numbers, integers mod p. 3. Linear equations over fieldsMatrices and elementary row operationsrank of a matrixSolutions of homogeneous and non homogeneous systems of linear equations and the connections between them.4. Vector spaces over fields Subspacesbases and dimensionscoordinates change of coordinate matrixrow rank as rank of a subspacesums, direct sums of subspaces and the dimension theorem. 5. MatricesMultipluicationinverse determinants: properties, Cramer's rule, adjoint and its use for finding the inverse.6. Linerar transformationsbasic properties, Kernel and Image of a linear trasformation and the dimension theorem.Representaion of linear transformations by matrices and the effect of change of basis.Eigenvectors and eigenvaluescharacteristic polynomialdiagonalization7. Inner product spacesstandard and nonstandard inner products on finite dimensional vector spaces and some examples of infinite dimensional inner product spaces.Cauchy Schwartz and the triangular inequalitiesGram Schmidt orthogonalization process.