Simple explanation of a vector space

[PainterGuy]

Hi, :)

I have some basic knowledge of matrices and vectors. mI understand some of their practical applications in real life. This idea of vector is entirely new to me. What is a vector in simpl terms, or to begin with? Please try to explain with some simple practical explanation. I would really appreciate this vector space teaching of yours. Many thanks.

 

[grief]

There are different ways to think of what vectors and vector spaces are. One way in which vectors are introduced in high school is "arrows" that have a magnitude and a direction. These arrows may be in R^n for any n. An arrow is a nice way of visualizing it, but really vectors may simply be understood as points in R^n. The reason you can move back and forth between arrows and points is that, given a point, you can form an arrow that goes from the origin to the given point. So a vector in R^n is really an n-tuple of numbers x1,...,xn.

But you probably know all this. Now there is this more abstract notion of a vector space. A vector space (let's say a vector space over the real numbers) is a collection of vectors, equipped with an operation of addition (you can add two vectors together) and an operation of scalar multiplication (you can multiply a vector by a real number). In addition, a vector space needs to satisfy a bunch of properties, such as commutativity and distributivity. You can find a complete list of these properties on wikipedia or in a textbook.

R^n is one example of a vector space. The vectors in this case are the points of R^n. The operation of addition is normal addition of vectors in R^n e.g. (1 2)+(3 4)=(4 6). And the scalar multiplication works by multiplying each coordinate e.g. 3*(1 2)=(3 6).

But with this more abstract notion of vector space, we can come up with other vector spaces as well. For example, the collection of functions from R to R is a vector space. We can add two functions together, and we can multiply functions by constants.

And the theory of linear algebra is the theory of vector spaces -- it works for R^n, but also for other vector spaces.

 

["pi"mp]

PainterGuy,

The above explanation is very nice and concise. Just to further emphasize, you can't geometrically imagine a vector as an arrow when you delve into vector spaces of dimension greater than three. But rather, you imagine a vector as a point in that vector space. If we're dealing with R^5, we must call a vector by a "five-tuple": (x1,x2,x3,x4,x5). Just like in R^2 (imagine the xy-plane), we need two "components" to effectively call a vector.

Hope that helps

 

[Deveno]

the most familiar examples of vector spaces, are the real plane, and R^3, which we can imagine as being an idealized version of the space we live in.

the usual way to identify a point in the plane or 3-space, is to give it "coordinates". and there is a 1-1 correspondence between a point's coordinates, and the usual way of describing "arrows" in terms of "unit vectors" such as i,j,k:

(x,y,z) <--> xi + yj + zk.

note the plus signs on the right-hand side. this is a different kind of addition than that between ordinary real numbers.

now, in the most general case, we have 2 structures, a field F, and the vector space V. without going into too much detail about what exactly a field is, a field is a set of numbers where one can do all the arithmetic you learned in high-school. normally, one is most interested in real or complex numbers (although other fields are sometimes used).

in some ways, the vector space V itself has LESS structure than the field, as we can (in general) only ADD (and subtract) vectors. what distinguishes the vector space V from just an additive structure, is the way that the field interacts with it: the scalar multiplication.

with scalar multiplication, we can "multiply" a vector by a field element (scalar). intuitively, you can think of this as "stretching", although in the general case of an arbitrary field, this does not always fit perfectly (what does it mean to "stretch by 2+2i"?). in any case, what we get from multiplying a vector by a scalar, is another vector.

the other "rules" (technically, axioms) for a vector space are to ensure that "scalar multiplication" and "vector addition" are compatible. a vector space is anything that satisfies these rules (usually there are 10 of them. some books have a different number).

 

[mathwonk]

vectors are any objects that can be added and subtracted and multiplied by numbers with all expected properties holding. e.g. if S is any set and V is the set of all real valued functions on S, then V is a typical (real) vector space.

Or we could take the subset W of V, consisting of all real valued functions on S such that the subset of S on which the function is non zero is finite. This gives actually every possible (real) vector space.

e.g. R^3 is the set of real valued functions on the set {1,2.3}. I.e.. a vector with foot at (0,0,0) and head at (x,y,z) corresponds to the function v such that v(1) = x, v(2) = y, and v(3) = z.