Vectors sometimes used as arrays, sometimes as concept of "change"

Question

As a first-year student at a small town college, I've been receiving conflicting information about what vectors (and matrices/tensors) are.

Sometimes, it seems that they are simply used as containers/arrays for multiple numbers in an ordered form. Other times, I've seen them being used to represent a concept of change in space, differentiating themselves from the concept of a point in space.

Is there a precise definition of what a vector (matrix/tensor) is that applies across all mathematical fields? Or am I correct in understanding that they are sometimes used to represent abstract concepts of change and other times they are used simply as arrays of numbers?

Is this just how things are? And instead of seeking a universal definition, should I accept that its use and concept may vary slightly depending on the context?

Answer 1

The formal definition of a vector is pretty open ended (a member of a vector space). At a very high level vector is a collection of mathematical objects, that obeys rules of addition and scalar multiplication.

A container of numbers isn't too bad. But the objects could be something like differential operators. And, they could be other vectors.

But then, what do these objects represent? They could be points on plane (or in space) and take Euclidean geometry into n-dimensions.

Physicists use them to model position, velocity, and acceleration of objects. And to represent forces acting on an object.

Since vectors obey rules of addition, and scalar multiplication -- they don't have to be the standard rules, they just have to follow some well-defined rule -- they form algebraic structures. Which opens up a world of just what is "Algebra."

The definition is pretty abstract, and the applications are manifold.

Answer 2

A vector space is a mathematical structure, and while in itself is quite informative, once you know a set is a vector space it doesn't mean your work is done, you can stop burning the midnight oil and hit the hay.

For example: there's an obvious vector space isomorphism between the space of $n-1 $ degree polynomials (call $\mathbb{R}[x] $) and $\mathbb{R}^n $. As far as the vector addition and scalar multiplication is concerned, they're the exact same. However, by treating them as the same it's possible to lose some of the additional properties of the polynomial space by considering the latter space in the stead of the former.

In the same way, you can have an isomorphism from $\mathbb{R}^n $ to its dual, which, as far as the vector space structure is concerned, it's the same vector space.

Similarly, when you consider the set of linear operators $Hom(V,W) $, or the set of vector space homorphisms from a vector space $V$ with dimension $n$ to a vector space $W$ with dimension $m$, there is an isomorphism from $Hom(V,W) $ to the set of $m \times n $ matrices. If you have $V = W $, you also have an isomorphism from the set of matrices to the set of bilinear forms $B: V \times V \rightarrow \mathbb{F} $ (where $\mathbb{F} $ is the underlying field).

What does this mean? Obviously these sets are not the same. Each of these sets can be used in different ways and have properties which have nothing to do with each other. To even think that that $Hom(V,V) $ is the same as the set of bilinear forms on $V$ would be a pretty grievous error. Each of these sets has additional structure that naturally give rise to certain properties that are pithy for one but inapplicable to the other.

It means, a vector space is a broad enough concept that two vector spaces which have the same dimension and are over the same field, can still have wildly different descriptions and functionality depending on the context, some of which only apply to certain types of vector spaces.

The value of the vector space as a metaphor is that it endows a broad, seemingly disparate group of sets with the ideas of angle (when you have an inner product), direction, and magnitude, even when it seems they have nothing to do with them.

Answer 3

The distinction you introduced between vectors as arrays (columns or rows), on the one hand, and vectors as change, on the other is well taken. I would like to comment on vectors as change, used in fields ranging from calculus, differential geometry, and dynamical systems to mathematical physics.

In this approach, a vector is thought of as an infinitesimal displacement that causes a change in another quantity, whose behavior is then studied. Physics courses are full of references to "bunches of small arrows" representing such change. In formalized language these are referred to as vector fields.

In mathematical frameworks based on the real number system there are no infinitesimals. For this reason a vector has to be reinterpreted as a quantity of real size and therefore not an infinitesimal size. This is a departure from the initial intuition but is extremely successful and provides an adequate foundation for the mathematical formalism.

Recently we have developed an approach to differential geometry via infinitesimal displacements that sticks closer to the original intuition of a vector as a tiny arrow. The approach relies an an enriched number system called the hyperreals; see this article.

From this point of view, the distinction between vector as array and vector as change is clear. They are not the same thing at all!