Do one dimensional vectors have a sign?

[Phrak]

Do one dimensional vectors have a sign? ()

 

[Werg22]

The answer is whatever you want it to be.

 

[Moo Of Doom]

Not in my book. Signs are a property of non-zero elements in an ordered field, not of elements in a vector space.

\mathbb{C}^1 is a one-dimensional vector space over \mathbb{C}, but its elements aren't split into positive and negative elements. You may also consider vector spaces over finite fields.

Even over ordered fields, one dimensional vectors probably should not be considered to have a sign. For example \mathbb{R}^1 is spanned by the vector (1). But there is an automorphism of \mathbb{R}^1 sending (1) to (-1). This automorphism doesn't preserve signs, so signs should probably not be property of vectors.

 

[chiro]

Do one dimensional vectors have a sign? ()

We call a one dimensional vector a scalar. If it is a set of the naturals then the sign is always positive. If it is over the integers we have positive sign past 0 and negative sign before 0. If it is over the reals the same thing applies.

 

[HallsofIvy]

Not necessarily. Certainly there exist an obvious isomorphism between a one dimensional vector space and (the additive group of) its underlying field of scalars, but it is not necessarily true that they are the same thing.

 

[Phrak]

I recall reading "vectors have magnitude, direction, but not sign."

u = 1 v = +v.

Clearly the vector +v is associated with a sign when the field is over integers or reals.

And just as clearly, none of the axioms of an abstract vector space, nor a vector defined as an ordered set of numbers like (1,2,3) in 3 dimensions supply this characteristic.

I see the answer to this question as no more than a preferred semmantical choice. Am I wrong?

 

[g_edgar]

Vectors have a direction. In one dimension, there are two directions only, and if you like you can call those directions + and - and refer to the direction as a "sign".