Introducing General Vector Spaces: Engaging Examples and Real-Life Applications

[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?

 

[Fantini]

Well, the definition of general vector spaces is an abstraction from the usual properties of $\mathbb{R}^n$. I'm not sure what kind of "real life" applications you could show. Almost everything usually is described using linear algebra, but you need additional information and structures for it to become interesting, like linear maps, decomposition into subspaces, etc. I don't think there'll be a vector space that's all useful just by existence.

 

[Ackbach]

The best real-life applications of vector spaces of which I am aware are differential equations, especially Sturm-Liouville type. Introductory Quantum Mechanics spends a lot of time solving the Schrödinger equation, which in its non-relativistic time-independent form, is a self-adjoint eigenvalue problem. You set up basis vectors for the eigenspace, and then you write down your solution with Fourier analysis - as a sum of eigenvectors.