Algebra with geometry II

physics

Algebra with geometry I

Classroom

The purpose of the course is to explain the basic notions of algebra such as eigenvalues and eingenvectors, bilinear and quadratic forms, scalar product.

The purpose of the course is to explain notions that appear in mathematics and physics throughout the entire period of studies. These abstract notions will be illustrated with various examples to make them maximally comprehensible and to demonstrate their usefulness in physics.

The foreseen workload is as follows:

1. Participation in classes 60 hours.

2. Homework and preparation for classes 30 hours.

3. Preparation for midterms and the exam 30 hours.

1. Systems of linear equations.

2. Eigenvectors and eigenvalues, decomposition into invariant subspaces.

3. Functions of a linear map.

4. Dual space, dual basis, adjoint of a linear map.

5. Bilinear and quadratic forms, the diagonalisation of a quadratic form, signature.

6. Scalar product, norm, orthogonal projection, volume.

7 Hermitean adjoint, spectral theorem, quadratic forms on Euclidean spaces.

8. Quadrics.

1. S. Zakrzewski, Algebra i geometria, Warsaw University publication.

2. P. Urbański, Algebra liniowa i geometria, Warsaw University publication.

After having completed the course students should:

a) be able to find eigenvectors and eigenvalues, compute functions of a matrix;

b) understand the notion of a bilinear and a quadratic form, know how to find their signature;

c) know the notion of a scalar product, an orthonormal basis, the orthogonal complement;

d) understand the notions of the Hermitean conjugate, selfadjoint and unitary operators;

e) know the spectral theorem for finite-dimensional complex case vector spaces;

f) be able to identify the type of a surface given by second order equations.

Midterms and written exam: computational part and basic theoretical part. Oral exam (optional): detailed theoretical part.

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