Program, Semester Week 9

Vector Spaces

When a set of vectors fulfills certain structural requirements of particular use, then we call the set a vector space. The structure of a vector space allows for convenient measures to be established, such as the concept known as a basis.

Vectors in a vector space are as such not different from the vectors we worked with last week, but this week we treat them in a much more generalized manner. For instance, a vector space does not necessarily have to contain vectors but can be a set of other mathematical objects such as matrices or functions. In particular we will study how linear (in)dependency can be understood in this more general setting.

From experience, students may find this generalized perspective slightly abstract and counter-intuitive, so we encourage you to invest extra effort in your studies this week and next week.

Key Terms

Vector space \(V\). Linear dependency vs. independency of vectors in a vector space. Basis. Dimension \(\mathrm{dim}(V)\). Subspace.

SymPy Demos

For this week we have a SymPy demo on General Vector Spaces.

Preparation and Syllabus

This week will cover Sections 9.1 through 9.3 from Chapter [09 - Vector spaces]. Note that Section 9.4 in this chapter is not a part of our syllabus.

Exercises

Exercises for Long Day.

Short Day is dedicated to Theme Exercise 3.