Are the definitions of vector in Linear Algebra and physics compatible?

[MManuel Abad]

I'm really confused. I study physics, and in Relativity we deal a lot with vectors and tensors.

I know an element x of Rⁿ is a vector, because Rⁿ is a vector space. That's fine.

Now, in physics, we define a (contra-) vector x^a (with 'a' an index running from 1 to n) as a n-tuple of quantities V^a that, under a coordinates transformation from x to x' given by:

x^{a}\rightarrow x'^{a}=x'^{a}\left(x^{1},x^{2},...,x^{n}\right)

transform as:

V^{a}\rightarrow V'^{a}=V^{b}\frac{\partial x'^{a}}{\partial x'^{b}}

where Einstein summation convention is used.

Everything is fine... 'till the question: ¿Are coordinates (cartesian, if you like) vectors?

A coordinate is an element of Rⁿ so, as the definition of Linear Algebra says, YES, coordinates are vectors.

But let's see if that is so in physics.

Let's consider the simplest coordinate transformation: that of adding a constant c^ato each coordinate:

x^{a}\rightarrow x'^{a}=x^{a}+c^{a}

Then, it's easy to see that, using physics definition, the coordinates x^a are not vectors (at least, not contra-vectors), 'cuz the expresion

\frac{\partial x'^{a}}{\partial x'^{b}}

gives 1 if a=b and 0 if a=/=b, so the definition would say:

x'^{a}=x^{a}

which, obviously, is not true.

¿So, what? Actually, I've never seen, in Relativity books, that the author calls the x^a contravectors. But even physicists call them a "Radio vector". What's going on? But I've seen that the differentials dx^a ARE contravectors (The physics definition works with the transformation I've used).

Could someone explain it to me? Are the definitions of vectors in Liner Algebra and physics contradictory?

 

[Stephen Tashi]

I'm not a an expert of the use of tensors in physics, but I'll make the following observations.

You have to be precise about what you mean by R_n. For example, if R_2 [/tex] simply means pairs of real numbers then there is no implied operation to add two members of R_2 [/tex] together. For example (a,b) might be the polar coordinates of a point on the plane, or it might be the cartesian coordinates of a point on the plane or it might be someone's height and weight. <br /> <br /> You example uses translation as a change of coordinates. In many contexts, a change of coordinates is assumed to be represented as multiplication by an invertible matrix. In 2 dimensional, cartesian coordinates for the plane, translation can't be represented as multiplication by a matrix. To use a matrix, you can represent a point (x,y) in the plane as (x,y,1) in "homogeneous" or "projective coordinates". Then<br /> <br /> \begin{bmatrix} 1 & 0 & c \\ 0 & 1 & c \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} = \begin{bmatrix} x + c \\ y + c \\ 1 \end{bmatrix}

 

[g_edgar]

The covectors constitute the dual space of the vectors. This tells mathematicians how to treat them when you change variables. Physicists have these weird formulas to use for that... Why? You'll have to ask a physicist.

 

[Rasalhague]

One way to look at it is, as Stephen suggests: Rⁿ, the set of tuples (finite lists) of n real numbers isn't itself a vector space; it's just a set. Only with the extra structure of scalar multiplication and componentwise addition do we construct a vector space using this set for its vectors. The usual way to express this idea is to say that the famous vector space is itself a tuple (Rⁿ,K,+,s), where K is the field of real numbers with the usual definitions of addition and multiplication, + is vector addition and s the scalar multiplication function (multiplying a vector by a real number). Because one kind of addition and one kind of scalar multiplication are so standard, and such natural choices, the common practice is to blur the distinction between the set of vectors and the extra structure that makes it into a vector space, and to speak as if the underlying set was itself the vector space.

Another way to look at it would be to say, okay, coordinates are vectors in Rⁿ (although, in this application, depending on the coordinate system, we don't necessarily use the full vector machinery), but, in physics, the unqualified word vector, by tradition, tends to refer to a vector of the tangent space associated with some point of a manifold representing space or spacetime. Vectors of other vector spaces tend to be given some other name or qualification, relating them to tangent vectors, such as covectors, tensors, etc.

 

[MManuel Abad]

Wow, thanks, that really helped. I'm thinking I'm going to study diff. geometry, and try to understand what's all of that: manifolds, tangent spaces and stuff. Thank you all :)