What is the Definition of a Vector in Mathematics?

[matteo16]

maybe I'm wrong but when i was following a lesson of mechanics by video lectures the teacher gave a very superficial definition of a vector as an arrow.
can i find the complete definition of a vector? maybe in maths?
otherwise I'm desappointed about this definition of a vector

 

[Diffy]

Hello. Definition of a Vector from Wolfram Mathematical encyclopedia

Bookmark this encyclopedia! It is a great resource, I would wager most people here would recommend it.

 

[HallsofIvy]

There are a number of related definitions!

In linear algebra, a vector is a member of a "linear vector space".

Perhaps the definition you want, for the kind of applications that appear in physics, is really the definition of a "tensor of order 0" (don't worry, we don't need the full definition). The basic idea is that, given a coordinate system, we can write a vector as set of numbers, components,(the number depending on the dimension) that change "homogeneously" when you change the coordinate system. Basically that means that the new components are each a sum of the old ones times numbers depending on two coordinate systems.

That might be a little "disappointing" also but think about this. If a "vector" has all zeros as its components in one coordinate system, then in ANY coordinate system its components are just those zeros times the "change of coordinate systems"- which must be 0. If a vector has all zero components in one coordinate system, then it has all zero components in any coordinate system!

Why is that important? Suppose you have an equation that says one vector is equal to another, in some coordinate system: A= B (A and B can both involve complicated operations as long as the result of those operations is a vector). That just says A-B= 0- it has all 0 components in that coordinate system. But then it has all 0 components in any basis and so A'- B'= 0 (A' and B' are the component forms of A and B in the new coordinate system) and A'= B'.If a vector equation is true in one coordinate system, then it is true in any coordinate system!.

Since coordinate systems are not "natural" but imposed when writing a mathematical model for a physical system, that makes vectors (and their more general cousins, tensors) the natural way to write physics equations.

An "arrow" is something of a simplification, but it holds the essense of the idea- If you set up different coordinate systems, you get different components- but the vector itself if still their, with the same length, pointing in the same direction- the vector itself is independent of the coordinate system.

 

[fyzikapan]

A vector is an object in a linear vector space. A linear vector space has rules for vector addition and scalar multiplication, with the restrictions that:

1. The space is closed (if you add two vectors or multiply a vector by a scalar, you get another element of the space)
2. Scalar multiplication is distributive in vectors and scalars
3. Scalar multiplication is associative
4. Addision is associative and commutative
5. There's a null vector
6. Every vector has an inverse

Notice that there's no requirement that vectors have a magnitude and direction. While arrows do indeed form a vector space, there's nothing restricting you from making a different vector space of something besides arrows.

 

[matteo16]

thank you for the definitions

but that i wanted to say is that in the video lesson of MIT that I've followed the teacher introduced vectors by a simple arrow
i know that it could semplify everything but i thougth that he would introduce vectors by some of definitions that you gave me or by this definition(i hope to write well because i don't speack english): a vector is each class of the partition of the oriented segments' set

 

[LukeD]

HoI: I'm assuming you mean rank-1 tensor since rank-0 tensors are isomorphic to scalars

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matteo16: The typical definition of a vector that you will see in most physics, engineering, and applied math courses is as an element of $$\mathds{R}^n$$. In other words, it is a point in n-dimensional space. The way it is usually used though is that you form it by subtracting two points in $$\mathds{R}^n$$ or in terms of other vectors that you've formed like this, so you may have something like $$\vec c = a - b$$.

Since a and b change in some predictable way when you change coordinates, you know that your vector c also changes in some predictable way. Any time you do translations, c stays the same. Any time you do rotations, c rotates by the same amount as the rotation. Any time you do reflections, c either stays the same or flips depending on how c is defined.

You also usually have some notion of "magnitude" on the vector defined. Usually, you have $$\textbf{c} = |\vec c| = ||\vec c||_2 = \sqrt{c_1^2 + c_2^2 + \cdots + c_n^2}$$. And with this, you can define the distance between two points in $$\mathds{R}^n$$ as $$d(a,b) = |\vec {a-b}|$$

Anyway, like I said, this is the typical working definition of a vector that you will encounter in any physical theory. The general definition of a vector was given above by fyzikapan, but as you can verify, the definition of a vector that you use in physical theories satisfies the properties of a vector in general.

 

[mathwonk]

i have always wanted to know what mit vectors are!

 

[LukeD]

i have always wanted to know what mit vectors are!

Oh man, so you're saying that these arrow things carry mit? I better stay away from them so as not to catch it

 

[Gib Z]

A simple, not very mathematically satisfying but sufficient for basic physics students purposes is: A quantity that has both a magnitude and a direction.

 

[matteo16]

ah ok thank you very much

 

[mathwonk]

there is no such thing as a vector. the name refers to a relation ship between the elements of a whole class of things.

a class of objects may be called vectors, with respect to a given collection of "numbers" if the numbers form a "field", and if for each pair of the "vectors" there is a way to add them to produce another one. also for each one, and each number. there is a way to multiply the vector by the number to get another "vector". also the usual expected operations of distributivity, commutativity, and inverses must hold.

if all this holds, the collection of objects is called a space of vectors over the field of numbers.

e.g. the collection of all arrows in the plane, emanating from a fixed origin, form a space of vectors with regard to the parallelogram law of adition,a nd the usual scaling rule for real number multiplication.

but the collection of all twice diferentiable real valued functions on the interval [a,b], also forms a space of vectors for addition of functions and constant multiplication of functions.so i guess arrows give an example of vectors, rather than the definition. i.e. vectors are any collection that behaves like arrows, i.e. just so you can add them and multiply them by scalars.