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As a freshman in a small town college. Ive been getting mixed signals to what vectors (and matrices/tensors) are.

Sometimes I get the feeling they are used just as containers/arrays for multiple numbers in an ordered form.

And sometimes I see them being used to denote a concept of change in space (whatever that means). And is used to distinguish itself from the concept of a point in space.


Is there like a very rigorous definition of what a vector (matrix/tensor) is that spans across all mathematical disciplines? Or am I right in that sometimes they are used to denote some abstract concept of change and other times they function just as an array of numbers?


Is this just the ways things are? And instead of trying to pursue a global definition, just understand that its use/concept is sometimes subtly different in different contexts?

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  • $\begingroup$ Well, fundamentally a vector is an element of a vector space, which is not very helpful. The question is: what are you representing with your vector space? If you're representing translations, then a vector is a translation; if you're representing arbitrary combinations of parameters, then a vector is an arbitrary combination of parameters. $\endgroup$ – Riccardo Orlando May 24 '16 at 22:23
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    $\begingroup$ As sometimes happens, your question indeed is its own answer: yes, context determines everything. "Vectors" can be many different things, depending on what we want. Yes, they have a few things in common, and it's smart and efficient to understand those commonalities, but "vectors" are soooo ubiquitous that the various contexts more powerfully influence the "sense" in a given situation than do the abstract commonalities. I think you understand that "formal math" does not explain all mathematical activities, even "pure math". Context. $\endgroup$ – paul garrett May 24 '16 at 22:37
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    $\begingroup$ As a general rule of thumb I like to use the words in the following way. Can I add to it other things like it and multiply it by a scalar? Yes - vector. No - tuple. $\endgroup$ – Dan Rust May 24 '16 at 22:47
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    $\begingroup$ You're welcome. Yes, as you might gather from the implied context (!) of my comment here, and many comments and "answers" throughout, I have come to appreciate a number of usually-suppressed things (in math), such as context. In math, perhaps a bit unlike more liberal-arts subjects, for many people there is a significant, and unhelpful, attraction to a weird, mute, non-verbal, fake-absolute context. It involves denial that there is context, for example. Many textbooks adopt such a viewpoint, and many teachers-of-math do also. Beware. This site does significantly better, I think! :) $\endgroup$ – paul garrett May 24 '16 at 22:49
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    $\begingroup$ @Alan by tuple I just mean an ordered set of elements with no additional structure. $(a,b,\mbox{donkey})$ is a $3$-tuple for instance. $\endgroup$ – Dan Rust May 26 '16 at 6:25
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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.

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  • $\begingroup$ thanks for your response. out of curiosity. I know that vectors obey a set of rules that define what a vector is. but do matrices and tensors also have a list of similar governing rules? $\endgroup$ – AlanSTACK May 24 '16 at 22:55
  • $\begingroup$ Yes, there are similar rules of addition and scalar multplication $\endgroup$ – Doug M May 24 '16 at 23:32
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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.

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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!

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