1. SylabUZ
2. Faculty of Mathematics, Computer Science and Econometrics
3. winter term 2019/2020
4. Mathematics - First-cycle studies leading to Bachelor's degree
5. Linear Algebra 2

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Linear Algebra 2 - course description

Aim of the course

The objective of the whole course (linear algebra 1 and 2) is to prepare participants to self-study of theoretical and practical problems involving methods of linear algebra. The aim of each student should be to master the material included in the recommended book.

Prerequisites

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Linear algebra 1.

Scope

Lecture
Systems of linear equations
1. Characteristic equation; eigenvectors; eigenvalues; examples and applications. (4h)
Jordan decomposition
1. Algebraic sum of linear subspaces; direct sum. (1h)
2. Nilpotent endomorphisms; Jordan blocks. Invariant subspaces of an endomorphism. (2h)
3. Jordan decomposition of an endomorphism; Jordan normal form. (2h)
Euclidean spaces
1. Cosine theorem — geometric definition of a scalar product; scalar product in coordinate spaces. (1h)
2. Formal definition of a scalar product; norm; Schwarz inequality; angle between two vectors, triangle inequality; parallelogram law. (2h)
3. Orthogonality: Pythagorean theorem, orthonormal basis.(1h)
4. Gram–Schmidt algorithm, existence of an orthonormal basis, expansion of a vector with respect to an orthogonal basis, orthogonal complement. (3h)
5. Isomorphic Euclidean spaces; canonical isomorphism between a Euclidean space and its dual. (1h)
6. Conjugate of a linear transformation; spectral theorem for self-adjoint operations.

7. Orthogonal transformations; decomposition of a space into minimal invariant subspaces: rotations, reflections. Canonical matrix of an orthogonal transformation. Orientation.(5h)
Bilinear forms
1. Multilinear forms: skew forms, symmetric forms. (1h)
2. Bilinear symmetric forms: matrix of a form with respect to a given frame. (1h)
3. Diagonalization of a bilinear symmetric form; Sylvester’s law. (2h)
4. Quadratic forms; polarization formula — the one-to-one correspondence between symmetric and quadratic forms. (1h)

Class
Systems of linear equations
1. Solving eigenvalue problems. (4h)
Jordan decomposition
1. Simple examples. Information on numerical packages. (2h)
Euclidean spaces
1. Finding the angle between vectors. Checking whether a given form is a scalar product (2h)
2. Finding an orthonormal basis by Gram–Schmidt orthogonalisation process. Gram’s determinant and its geometrical interpretation. (5h)
3. Class test
4. Diagonalisation of simple self-adjoint transformations. (4h)
5. Classification of orthogonal transformations in dimensions 2 and 3. Composition of orthogonal transformations. Reduction of orthogonal matrices to their canonical forms – examples. (5h)
Bilinear forms
1. Matrix of a bilinear form. Decompsition of a form into skew and symmetric parts. (1h)
2. Diagonalization of bilinear forms (quadratic forms). (2h)

Teaching methods

Traditional lecturing, solving problems under the supervision of the instructor.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

1. Preparation of the students and their active participation is assessed during each class by their instructor.
2. Class tests with problems of diverse difficulty helping to assess whether a student achieved minimal outcomes.
3. Written examination: It consists of around 18 problems. Each problem consists of 2 or 3 statements. To solve a problem, one has only to decide whether the statements are true or false. For some of them, however, explanations are demanded.

Final grade = 0.4 x class grade + 0,6 x exam grade. In order to be allowed to take the exam a student has to have a positive class grade. In order to pass the exam a student has to have a positive exam grade

Recommended reading

1. Strang, Gilbert, Linear Algebra and Its Applications, Cengage Learning, 2005.

Further reading

1. G. Birkhoff, S. Mac Lane, A Survey of Modern Algebra, A.K. Peters, 1997.

Notes

Modified by dr Robert Dylewski, prof. UZ (last modification: 19-09-2019 13:36)