Agenda, Semester Week 7

Matrix Algebra and Determinants

This week concerns vectors and matrices. A matrix was introduced last week as a convenient notation form when solving systems of linear equations - this week we will show that a matrix can be considered as a mathematical object in its own right. We will study how matrices can be multiplied together and how arithmetic operations and properties can be defined on such objects. Also, the important concept of linear dependence vs. linear independence will be mentioned, a topic that will become immensely important in the following weeks.

This week will also introduce the concept of determinants, a property of square matrices that often is useful. As we will see, the determinant can be used to investigate easily whether vectors are linearly independent or not.

Remark: Python Test

This Long Day the Python test for Python module 1 will take place. It will be based on the Jupyter Notebook released last week and will be carried out in Möbius, so bring your laptop. You must be present in the exercise area during the test (it cannot be done from home). It opens at 15:30 and closes at 17:00. As aids you may use notes and books, all digital material on the course website as well as your chosen software for Python/SymPy computations on your laptop - you may not access any other electronic or online tools during the test.

Key Terms

Row vectors and column vectors. Matrices. Square matrices, the zero matrix \(\mathbf 0\), and the identity matrix \(\mathbf I_n\). Triangular matrices and diagonal matrices. Matrix arithmetics, including matrix multiplication. Transposing a matrix. The inverse matrix. Linear dependence and linear indepedence. Submatrices. Determinants \(\mathrm{det}\).

Preparation and Syllabus

This week will cover Sections 7.1 through 7.3 until and including Equation (7-10) from Chapter 7 as well as Sections 8.1 and 8.2 from Chapter 8 in [the textbook].

Exercises

Exercises for Long Day.

Exercises for Short Day.