- #1
Phrak
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Do one dimensional vectors have a sign? (
)
)Do one dimensional vectors have a sign?
- Context: High School
- Thread starter Phrak
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- One dimensional Sign Vectors
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Discussion Overview
The discussion centers on whether one-dimensional vectors possess a sign, exploring the implications of this concept in various mathematical contexts, including vector spaces and fields. Participants examine the definitions and properties of vectors, particularly in relation to their scalar fields.
Discussion Character
- Debate/contested
- Conceptual clarification
Main Points Raised
- Some participants suggest that the sign of a one-dimensional vector is subjective and can be defined as desired.
- One participant argues that signs are properties of non-zero elements in an ordered field, not of elements in a vector space, citing examples from complex and finite fields.
- Another participant notes that while one-dimensional vectors can be associated with scalars, they do not inherently possess signs, emphasizing the distinction between vector spaces and their underlying fields.
- One contribution mentions that in specific fields, such as the naturals or reals, the sign can be defined based on the value of the scalar, but this is not universally applicable.
- A participant recalls a perspective that vectors have magnitude and direction but not sign, suggesting that the association of sign may be a semantic choice rather than a mathematical necessity.
- Another viewpoint posits that in one dimension, there are only two directions, which can be labeled as positive and negative, thus allowing for a notion of sign.
Areas of Agreement / Disagreement
Participants express differing views on whether one-dimensional vectors can be said to have a sign, with no consensus reached on the matter. Some argue for the existence of signs based on the scalar field, while others contend that signs are not intrinsic to vectors themselves.
Contextual Notes
The discussion highlights the dependence on definitions of vector spaces and fields, as well as the implications of dimensionality on the concept of sign. The relationship between vectors and their underlying scalars remains a point of contention.
- #2
Werg22
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The answer is whatever you want it to be.
- #3
Moo Of Doom
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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.
\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.
- #4
chiro
Homework Helper
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Phrak said: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.
- #5
HallsofIvy
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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.
- #6
Phrak
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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?
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?
Last edited:
- #7
g_edgar
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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".
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