Figure of speech to help explaining the mathematical concept of "vector spaces"

Question

I am teaching math in a community college and have to explain the idea of Vector Space that is an abstract concept but ubiquitous in high level math. I would like to explain it using a certain figure of speech but I don't know what it is.

In the most simplistic terms, in mathematics, a "space" means a collection, set or aggregate of identical objects. A "vector," as most students visualize it, is an arrow that has direction and magnitude. However, a vector space does not mean "a collection of arrows," but rather a collection of numerical objects that have the same properties as a vector. (Those properties are add-ability, scalability but not multiplicability. For example, the temperature readings have these properties.)

Is there any figure of speech to describe it? Is it metaphor or simile? I would love to get from you the experts couple of examples in daily use.

Answer 1

Renowned physicist Richard Feynman, when lecturing to a more general audience than physicists and mathematicians, often used the word "arrow" in lieu of "vector".

For example, he uses it in the book "QED" which is a cleaned up set of transcripts from a four part lecture series to the general public.

And, while I don't have the book at hand, I recall that he used a phrase quite close to a "collection of arrows" to describe path integrals and vector spaces.

I believe that he also did so in some of the Feynman Lectures on Physics.

Often what one does in a "lies to children" pedagogy is to use a strictly inaccurate term like "collection of arrows" and then either footnote in writing or orally caveat in speech, that while you are using this term, your metaphor isn't strictly accurate. Then you explain why this is the case.

Answer 2

Although I do agree with user Peter Shor concerning his claim that the OP's question doesn't belong to the present site, I'll still make the following answer as it could add something interesting to the subject.

In the concepts of everyday life there seems to be nothing tangible that will truly appear as having the nature of a vector space. The closest we might come to in the way of producing a vector space like entity could be found in the phenomenon of the opposition to current flow in an electrical circuit, where this opposition , called the impedance, is an ordered pair having for first element the resistance and for second element the reactance.
It is, however, grossly inadequate as a true vector space: there does exist positive and negative reactance (for equal magnitudes they cancel one another as do a vector and its inverse) but negative resistance is not a concept belonging to electricity; none of the would be vectors can have an inverse, multiplication by a negative scalar has no meaning at all. Except for this shortcoming we do have a componentwise addition and a multiplication by positive scalars for entities with the characteristic of being justified only by two components which are not of the same sort and not involving space.

Therefore, outside of the current examples of physics, where the components are homogeneous, except for relativity when to three components of space a time component is added to justify 4-dimensional space-time vectors, there seems to be nothing to talk to us of vectors in real life. On top of that, this example from electricity is still way too far removed from the realm of everyday life as the qualitative appreciation of the two effects still requires some understanding of many abstract concepts of physics. If we were nevertheless to refer to this phenomenon, we could then talk of a simile since there is no archetype of a vector space but only an approaching, crude model.

• A vector space is a system somewhat like an electrical series circuit with various points of opposition to current flow in it, there being at each of those points a variable resistance and a variable reactance that combine to produce a given impedance changing with time; the set of these oppositions to current flow under the operations of addition and multiplication by positive scalars has much of the nature of a vector space.

The combination of series impedances

                     

Answer 3

n-dimensional Euclidean space, for example, is a vector space in which the ordered n-tuples (x1, x2, x3, …, xn), a.k.a. the coordinates of points in n-dimensional space, play the role of vectors. That is, they satisfy the properties of vectors in the usual sense if one considers the tail to start at the origin of coordinates and the tip to end at the coordinate (x1, x2, x3, …, xn). More broadly, a vector space is any set of elements whose members have the same properties as vectors. For n-dimensional Euclidean space, those elements are the n-tuples (x1, x2, x3, …, xn). Other vector spaces, i.e., sets, will have their own terminology for the members of the set. Another example is the set of polynomials of degree n, which compose a vector space of dimension n + 1. There is no one single name or figure of speech for the elements of a vector space.