Linear algebra and geometry I 1000-111bGA1b

1. Systems of linear equations. General solution. Matrices. Elementary operations on rows. Reduced echelon form. Application to equation systems. (1 lecture)
2. Fields. The field of complex numbers. The trigonometric form of complex numbers. Roots of polynomials. The fundamental theorem of algebra (without proof). Roots of unity. The fields Z_p. (2 lectures).
3. Linear (vector) spaces. Subspaces. Linear combinations, spaces spanned by a set of vectors. Linearly independent systems. The Steinitz exchange theorem. Bases. The existence of a basis. The dimension of a linear space. Coordinates of a vector in a basis. The rank of a matrix. The Kronecker-Capelli theorem. Subspace representation by systems of linear equations. Intersection and algebraic sum of subspaces, dimension of the sum. Internal direct sum. (4 lectures).
4. Linear mappings. Operations on linear mappings (sum, multiplication by a scalar, composition), the space of linear mappings L(V, W). Homotheties, projections and parallel symmetries. Defining a mapping by its values on a basis. Kernel and image of a mapping. Monomorphisms, epimorphisms, isomorphisms. Every n-dimensional linear space over K is isomorphic to K^n. Dimension of a space as determined by the dimension of the kernel and image of a linear mapping. Matrix of a linear mapping. Algebra of matrices. Invertible matrices. Linear forms, dual spaces. Dual bases, coordinates of a form in a dual base, isomorphism of a finite dimensional space with its dual. Dual maps, their matrices in dual bases. (5 lectures).
5. Determinants. Properties of determinants. Elementary operations in the computation of a determinant. Laplace expansions. Cauchy's theorem on determinant multiplication. Application of determinants, relations to order and invertibility of matrices. Cramer formulas for the solution of a system of n linear equations. The permutation formula for determinants. (3 lectures).