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Week 7, Long Day: Vector Spaces

Today we will work with vector spaces in more general forms. Vector spaces are sets of widely different mathematical objects that have some decisive properties in common. You will likely experience that, regardless of whether the exercises are about polynomials, geometric vectors, matrices or tuples of numbers, the solution methods are structurally identical. Again today the central concepts are bases and coordinates.

Today’s Key Concepts

Spans or spannings. Linear dependence and independence. Basis and coordinates. Dimension. Coordinate matrices. Subspaces.

Preparation and Syllabus

Today’s eNote is eNote 11 General Vector Spaces.

Maple Syllabus

Many of our already learned Maple commands apply for today’s topics as well. In particular, commands regarding polynomials might be of smart use, which today’s Maple demo will demonstrate, but today’s exercises can be solved without theses commands.

Today’s Maple Demo is: VectorSpaces.

When a matrix is so large that Maple won’t show the output, then use interface(rtablesize=[insert a number]) (a useful command for Theme2).

Activity Program $ $

  • 10.00 – 12.00: Lecture (aud. 42, b. 303A) (link to streaming)
  • 12.30 – 17.00: Group exercises in the study areas (b. 302, bottom floor)
  • 13.00 – 16.00: Your teachers are present in the study areas
  • 15.30 – 16.00: The teachers will answer conceptual questions to the homework

Today’s Group Exercises:

  1. Dimension and the Zero Vector in Different Vector Spaces
  2. Linear Dependence or Independence
  3. Bases and Coordinates
  4. Monomial bases
  5. Subspaces
  6. Bases for Subspaces
  7. Vectors Within and Outside a Subspace
  8. Bases for Spans

Theme Exercise 2:
On Friday this week there is no lecture but you should instead work with Theme Exercise 2. The material for the Theme Exercise will be uploaded to Learn at 17:00 today. The Theme test in Möbius opens for answers on Friday between 14:00-17:00. Your Theme result will be included in your Homework portfolio grade. Make sure that you are in a group of 4–6 students and begin working on the Theme exercise early.