Linear algebra and geometry I

obligatory courses

The course GAL I is concerned with the study of systems of linear equations over fields. We define a field and study properties of fields of real and

complex numbers. The solution set of a system of linear equations will be endowed with the structure of a linear space. Matrices representing linear equations and linear transformations will be the main tool.

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).

Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0387982590

Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978-0321287137

Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM),

ISBN 978-0898714548 . Available online at http://www.matrixanalysis.com/DownloadChapters.html

Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3

Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International

Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall

1. They know the notions of a system of linear equations, its solution and its matrix. They know elementary operations on matrices and can apply them to find the solution of a system of linear equations using Gauss elimination.

2. They know the notion of a field and basic examples of fields. They know the field of complex numbers. They can present complex numbers in trigonometric form and use this presentation for calculating powers of complex numbers.

3. They know the notion of a linear space and examples of linear spaces. They can verify whether a given subset of a linear space is its subspace.

4. They know the notions of linear independence, a basis and the dimension of a linear space. They know basic properties of bases. They can find bases of a linear space. They can find a presentation of a subspace of a coordinate linear space as the solution space of a system of linear equations.

5. They know the notion of an algebraic sum of subspaces of a linear space and a direct sum of subspaces. They know the formula for the dimension of an algebraic sum of subspaces. They can verify whether a linear space is a direct sum of its given subspaces.

6. They know the notion of linear mapping and examples of linear mappings. They can find a formula for a linear mapping that is defined by its values on a given basis.

7. They know the notions of the image and the kernel of a linear mapping and the notions of a monomorphism, epimorphism and isomorphism. They know and can apply the formula relating the dimensions of the kernel, the image and the domain of a given linear mapping.

8. They know basic operations on matrices (addition and multiplication) and their properties. They know the notions of an invertible matrix and of the inverse matrix. They can check whether a matrix is invertible and, if so, find its inverse.

9. They know the notion of the matrix of a linear mapping and know how this matrix depends on the choice of bases. They can calculate the matrix of a linear mapping.

10. They know the notion of a linear functional and of the dual space. They can find the coordinates of a linear functional in the dual basis.

11. They know the notion of the rank of a matrix and its relationship with the rank of the linear mapping. They can apply the rank when solving systems of linear equations and checking the invertibility of matrices.

12. They know two definitions of the determinant: the inductive definition and the definition via permutations. They know basic properties of the determinant and can apply these properties for calculation of the rank of a matrix, the inverse matrix and for solving systems of linear equations. They know the geometric meaning of the determinant of real-valued matrices.

The credit will be based on the students's performance on the exercise sessions and two midterm exams.Marks will be given according to points, which can be obtained in the following way.

a) 200 points for tests - two tests in a semester, (80 + 120 points).

b) 80 points for classes, including at least 50 points for homework.

c) 20 points for online tests on Moodle (https://moodle.mimuw.edu.pl/course/view.php?id=531).