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 quickly determine whether vectors are linearly independent or not.

Remark: Python module 1

This Long Day a Jupyter Notebook will be released on the course’s Learn page at 15:30. This file will guide you through the use and syntax of important and useful commands in the Python package Sympy. Sympy will be your mathematical CAS tool in Mathematics 1b and in future courses at DTU. You will need an editor installed that can run Python (such as VS Code). Next week on Long Day a Python test will be carried out in the material from this Jupyter Notebook.

Key Terms

Long Day will cover:

• Row vectors and column vectors.

• Matrices as objects.

• Square matrices, the zero matrix \(\mathbf 0\), and the identity matrix \(\mathbf I_n\).

• Triangular matrices and diagonal matrices.

• Matrix arithmetics, including matrix multiplication.

• The transpose.

• The inverse matrix.

Short Day will cover:

• Linear dependence and linear independence.

• 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.