Dimension of a Set: Definition & Explanation

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Discussion Overview

The discussion centers on the concept of dimension, particularly in relation to sets and vector spaces, with a focus on the dimension of sets of matrices. Participants explore definitions, applications, and the necessary structures for discussing dimension in various contexts.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants suggest that the dimension of a set refers to the dimension of the space spanned by that set.
  • Others argue that dimension is defined by the number of coordinates needed to specify every element of a set, providing examples from R^3 and the set of 2x2 matrices.
  • One participant asserts that dimension cannot be discussed without a vector space context, emphasizing that arbitrary sets do not have a defined dimension.
  • Another participant expresses that "space" and "set" can be used interchangeably, but acknowledges that dimension applies only to linear sets or spaces.
  • A later reply introduces the idea that different notions of dimension exist beyond linear dimensions, referencing Poincaré's work on the topic.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the applicability of the term "dimension" to arbitrary sets versus structured sets like vector spaces. There is no consensus on a singular definition or understanding of dimension across all contexts discussed.

Contextual Notes

Some participants note that a structure is necessary to define dimension, particularly in the context of matrices, and mention that various definitions of dimension exist, including those from topology.

Who May Find This Useful

This discussion may be useful for individuals interested in mathematical definitions of dimension, particularly in the context of vector spaces and matrices, as well as those exploring different interpretations of dimension in various mathematical frameworks.

Gear300
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I'm familiar with the notion of the dimension of a vector space. Sometime earlier though, I ran into something asking for the dimension of a set of matrices. In general context, what is meant by the dimension of a set?
 
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I would think it's asking for the dimension of the space spanned by the set.
 
Its the number of coordinates you need to specify every element of the set. In R^3 each element is given uniquely by 3 real numbers so the dimension of R^3 is 3. The dimension of a sphere is two because you need only two angles to describe a point on the sphere. The dimsensio of the set of 2x2 matrices is 4 because you need 4 (ordered) real numbers to describe each matrice. In general the dimension of the set of all nxn matrices is n^2.

The definition of demension for a vector is the number of elements in any basis for that space.
 
Gear300 said:
I'm familiar with the notion of the dimension of a vector space. Sometime earlier though, I ran into something asking for the dimension of a set of matrices. In general context, what is meant by the dimension of a set?
NOTHING is meant by the dimension of a set! You must have a vector space to talk about dimension. And since you can add matrices and multiply matrices by numbers, the set of all m by n matrices is a vector space. But it makes no sense at all to talk about the "dimension" of an arbitrary set.
 
My understanding is that the term "space" and "set" are the same and may be used interchangeably.

The definition of a dimension of a set is applicable only to a linear set/space.

As the matrices form a linear operator set/space, the definition of dimension applies as in Deluks917.
 
Thanks for clarifying things...I was sort of confused when I started referring to the dimension of sets without structure.
 
the concept of dimension is discussed for lay persons in an essay by poincare in his collection, science and hypothesis, i believe. some structure is needed to define dimension. so in the case of matrices it would presumably be the structure imposed by viewing their entries as euclidean coordinates. there are more notions of dimension than just linear dimension though. e.g. a circle spans the plane linearly but has intrinsic dimension one according to poincare's criterion that a circle can be separated by removing a finite set of points. some topology at least is needed to discuss "separation".
 

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