What are some real-life applications of general vector spaces?

[matqkks]

Students familiar with Euclidean space find the introduction of general vectors spaces pretty boring and abstract particularly when describing vector spaces such as set of polynomials or set of continuous functions. Is there a tangible way to introduce this? Are there examples which will have a real impact? I would like to introduce this in an engaging manner to introductory students. Are there any real life applications of general vector spaces?

 

[jambaugh]

I try to give my students a "dynamic" definition. Basically, a vector space is an abelian Lie group. Of course I don't say that out loud but I start with arrows in Euclidean space as "displacement vectors", then a position vector is def'ed i.t.o. displacements of an origin point. Addition becomes composition of action.

I use the riddle "I walk South 1 mile, East one mile, and then North one mile and am back where I started. Where am I?" to show that we must be careful to confirm the basic properties (and hint at more general constructs).

Then when I jump to more abstract vectors we look to the actions involved in the algebra methods, e.g. solving systems of equations evokes the act of adding functions. I try to convince them that viewing appropriate abstract constructs as vectors gives them a set of "power tools" for attacking higher order problems. I show for example the connection between solving a linear ODE, solving a linear system and solving a simple Lin. eqn. Ax=b.

 

[Vaedoris]

Students familiar with Euclidean space find the introduction of general vectors spaces pretty boring and abstract particularly when describing vector spaces such as set of polynomials or set of continuous functions. Is there a tangible way to introduce this? Are there examples which will have a real impact? I would like to introduce this in an engaging manner to introductory students. Are there any real life applications of general vector spaces?

Are you talking about teaching inner product spaces?

If this is the case, it was mind boggling to me when I first learned that Gram-Schmidt Process and orthogonal projection can be be applied on vector spaces such as the underlying space of Fourier series, the set of polynomials, etc. Of course, understanding inner product itself is important.

I think only those who can see the big picture of how general inner product space is, can really get the excitement.

P.S.: I'm just a student.