Introduction to Vectors

matqkks

What is a good way to introduce vectors on a linear algebra course so that students are motivated from the start? I need an opening which will have a real impact. Are there any motivating examples?

Simon Bridge

Bribery is a proven motivator...

If your students are already solving linear puzzles, you can usually get a good result by moving to a puzzle that must use at least two dimensions to solve. The main trick is to avoid the dryer math-first approach where you define vectors and how they transform and then provide examples. It's a robust approach with a long history of boring students as I'm sure you remember.

Trouble is, the other way means you need to let the students lead you.

chiro

For linear algebra, it's probably a good idea to say that in normal Euclidean space each component of the vector is completely independent of the other vectors.

The reason why understanding independence is because the minimum number of independent things in a system describes the simplest way to describe such a system. Because of this, linear algebra is essential because it allows us to figure this out in indirect ways.

Also explain that linear algebra is directly related to the geometry of R^n because in R^n all axis are orthogonal and independent, which means that we can convert between geometry and algebra through linear algebra.

What this translates into is that if we have systems of n variables, then we can treat something as n-dimension vector if they are independent.

The other thing about vectors is that you can add lots of vectors together and get one output vector, which provides motivation for things like physics when we consider net result vectors of force, acceleration, momentum and so on.

Then move on to say that linear algebra is a systematic way of looking at things that behave like arrows: (i.e. vectors). Then you can explain why we look at spanning, basis, dimension and that kind of thing.

brimacki

Give them the following scenario:

"To get home Bob walks 13 miles, then 5 miles, then 2 miles"

vs.

"To get home Bob walks 13 miles north, then 5 miles east, then 2 miles south"

(I know it sounds ridiculous, change up the magnitudes if you need to...)