# Abstract Vector Spaces

I first introduce the vector along the lines 'something with magnitude and direction'. Later on the definition of a vector becomes generic - 'an element of a vector space'.
Euclidean spaces (n=2 and n=3) are something we can all visualize. However when describing other vector spaces such as set of polynomials or set of continuous functions all this stuff becomes abstract and many finding this severely boring. Are there other vector spaces which students will find attractive? Are there any real life examples of vector spaces which would be a good hook?
Thanks in advance for any replies.

You didn't say who you are teaching to?

Many schoolboys found Euclid intensely boring in the past, although much of the mathematical thought of the day was founded there and needed to progress in many technical disciplines.

The modern edifice (which I assume is somewhere in the course) founded on groups / mappings / linear algebra / forms etc is equally important to modern technical disciplines. And equally tedious for most to slog through.

All I think you can do is to mix applications with theory, not too much of one or the other. Individuals will differ as to which application will motivate.

go well

jambaugh
Gold Member
When I last taught Linear Algebra I basically (without using the terms) defined a vector space as an abelian Lie group. I then explained the example of the group of point translations (again without using the term "group") This I found helped the students understand e.g. why Euclidean vectors could be positioned anywhere and how the visualized addition rule worked. "First slide this way, then slide that way, the net motion of an arbitrary point is then the diagonal of the head-to-tail connection of the arrows".

In fact I started the first day's lecture with the old riddle "I walk south 1 mile, east one mile then north one mile and am back where I started. Where am I?"

The riddle's answer points out the importance of the geometry in making addition of translations commute hence that they are vectors. We go over each of the properties in this context: existence of an inverse (negative) and identity (zero) and scaling of actions (scalar product).

I found it helps to find a juicy application when covering e.g. abstract spaces. For example with the polynomials or function spaces I might mention "getting rich" playing the stock market and considering how one might construct indicators by adding or averaging stock prices as functions of time.

I also use the "power tool" analogy when we hit some mathematics with less obvious or less immediate utility. "We're playing with our power tools to learn how they work so we can use them to good effect when we need them".

Finally in teaching any subject it is important to show your own enthusiasm for the subject. Enthusiasm and its opposite are equally infectious.