Geometry with linear algebra

obligatory courses

Classroom

Basic notions and methods of linear algebra for computer scientists: fundamental algebraic structures, polynomials, linear spaces, systems of linear equations, Gaussian elimination, linear mappings and functionals, euclidean and unitary spaces, hermitian and symmetric forms.

1. Groups. Fields. Complex numbers, trigonometric form, De Moivre's formula, roots of unity, roots of a complex number.

2. Polynomials, fundamental theorem of algebra (without proof).

3. Matrices over field, operations on matrices.

4. Linear spaces over fields. Linear subspaces, linear independence, basis and dimension of a linear space. Examples of bases. Intersection, sum, and direct sum of subspaces.

5. Image, kernel and rank of a matrix. Invertible matrices.

6. Systems of linear equations. Kronecker - Capelli theorem. Description of solution set. Gaussian elimination.

7. Determinants and their properties. Cramer's rule.

8. Linear mappings and functionals. Matrix of a linear map. Image, kernel and rank of a linear map and matrix. Isomorphism of linear spaces.

9. Dual spaces and dual bases. Change of basis. Relationship to linear mappings.

10. Matrix similarity. Eigenvalues and eigenvectors of matrices and linear maps. Characteristic polynomial. Jordan normal form and Jordan decomposition theorem.

11. Euclidean and unitary spaces. Scalar product and euclidean norm of vector,

angle between vectors. Orthogonal and orthonormal bases, Parseval's identity. Gram - Schmidt orthogonalization. Orthogonal complement and orthogonal decomposition. Isometries and orthogonal / unitary matrices.

12. Hermitian and symmetric forms. Congruent matrices. Diagonalization of symmetric and hermitian matrices. Sylvester's criterion.

1. G. Strang, Linear algebra and its applications, Academic Press, 1976.

2. Lay, David C., Linear Algebra and Its Applications (3rd ed.), Addison Wesley, 2005

3. H. D. Ikramov, Linear algebra : problems book (transl. from Russian by Oleg Efimov), Mir, 1983

Knowledge:

1. Understands the notion of field, and fields of real and complex numbers in particular.

2. Knows the notion of matrix and understands operations on matrices.

3. Understands the notions of linear space, linear independence of vectors, basis and dimension. Is familiar with examples of linear spaces and their bases.

4. Understands the notions of image, kernel and rank of matrix. Is familiar with methods of finding these spaces. Can use these notions to describe the solution set of a system of linear equations.

5. Understands the notions of linear functional, dual space and dual basis.

6. Is familiar with methods of solving systems of linear equations of arbitrary. size.

7. Understands what is a linear map and its matrix. Understands the notion of isomorphism of linear spaces.

8. Knows the notions of eigenvalue and eigenvector.

9. Knows the definition and properties of scalar product. Understands the notions of euclidean / unitary space, orthogonality. Understands the connection between orthogonal projects and optimal approximation.

10. Understands hermitian and symmetric forms.

Skills:

1. Can perform operations on matrices and calculate image, kernel and rank of a matrix

2. Can solve a system of linear equations

3. Can calculate eigenvalues and eigenvectors of matrices and linear maps

4. Can use and apply notions and theorems of linear algebra on abstract level as well as in relation to concrete examples.

Compenetences:

1. Understands the significance of linear algebraic structures as a fundamental tool for creating and analysing complex mathematical models, including ones which describe the real world.

In order to obtain a positive grade students are required to obtain a certain minimal number of points granted for homework assignments, quizes, mid-term exams and exercise classes, as well as the final egzam.

Final egzamination is in written form.