Dimension Definition and 906 Threads

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.
In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. 10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space.
The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.

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  1. G

    MHB Dimension & Subspace of $\mathbb{R}^3$

    Check whether the following are subspaces of $\mathbb{R}^3$ and if they're find their dimension. (a) x = 0, (b) x+y = 0, (c) x+y+z = 0, (d) x = y, (e) x = y= z, and (f) x = y or x = z. (a) Let $S = \left\{(x, y, z) \in \mathbb{R}^3:x = 0 \right\}$. I want to check whether $S$ is subspace of...
  2. jk22

    How does Coulomb's law change in 3+1 dimensional spacetime?

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  3. G

    Linear algebra: Prove that the set is a subspace

    Homework Statement Let U is the set of all commuting matrices with matrix A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}. Prove that U is the subspace of \mathbb{M_{3\times 3}} (space of matrices 3\times 3). Check if it contains span\{I,A,A^2,...\}. Find the...
  4. Y

    R^(1/2) Question, half dimension space meaning?

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  5. P

    Difference between direction and dimension

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  6. G

    Quantum potential problem -- Particle confined in 1 dimension

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  7. Ez4u2cit

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  8. N

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  9. S

    Why Do the Terms in the Lagrangian Action Have the Same Dimension?

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  10. Safinaz

    Dimension of a partial decay width

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  11. T

    Is the Dimension of Phase Space for a Rigid Diatomic Molecule Actually 12?

    Dimension of Phase Space of a rigid diatomic molecule is 10. Shouldn't it be 12?
  12. M

    Time as a dimension, can spatial length be applied to it?

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  13. G

    Find a basis and dimension of a vector space

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  14. Y

    F90 Subroutine that accepts any array of any dimension

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  15. S

    Linear Transformation (Image, Kernel, Basis, Dimension)

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  16. Jaami M.

    Exploring the 4th Dimension: A Discussion

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  17. 24forChromium

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  18. Rickswan

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  19. S

    Dimension of SO(n) and its generators

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  20. M

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  21. ichabodgrant

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  22. I

    Understanding proof for theorem about dimension of kernel

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  23. M

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  24. M

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  25. B

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  26. Dimitri655

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  27. Dimitri655

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  28. kostoglotov

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  29. W

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  30. TheMathNoob

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  31. M

    Dimensional Analysis: Solving E = (1/2) mv Equation

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  32. Imran Makhdoom

    Exploring the Fourth Dimension: Can You Understand It?

    What is the fourth dimension can anybody explain it to me ?
  33. D

    GR: Signature & Dimension of Embedding Space

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  34. TachyonRSC

    B Time Travel - Does It Create a New Dimension?

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  35. praveena

    Which dimension does the black hole belongs?

    I want to know which dimension does the black hole belongs? Can anyone say which force is responsible for the absorption? In case, if the black holes absorbs everything then were the things might gone?Is that everything becomes invisible or just blast into pieces?
  36. Aner

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  37. N

    What is the 4th Dimension - 3D World + Time or 4 Axis?

    Some people say that it's the 3D world we live in + time but others say that it's 4 dimensional as in it as new axis. Which one is true?
  38. PerpStudent

    Dimension of the stress energy tensor

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  39. popopopd

    Solution space of nth order linear ODE, n dimension Vector Space

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  40. S

    Like spacetime -- can sound be taken as a fifth dimension?

    Like spacetime , can sound be taken as a fifth dimension?
  41. Bollini Mouli

    Understand 5th Dimension: Get Help Understanding 4th Dimension

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  42. Nemoto

    Is a photon of dimension zero?

    I have read that the elementary particles of the Standard Model have dimensions 0. Is this the case or not? I have read on this site relevant answers to similar questions, but have not found them to be very clear. If it is the case, surely much of quantum weirdness is thereby explained: after...
  43. nikosbak

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  44. hideelo

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  45. at94official

    Is Velocity Zero if Displacement is Zero?

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  46. K

    Holographic Universe. 2D Universe = Matrix?

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  47. T

    Dimension change of opening through concrete slab expansion

    Hi, apologies for the length of this, I'm hoping as a group you knowledgeable established or future Engineers and Physicists can help me out with a problem. I have been having a discussion with a colleague about expansion of a 6m x 7m integrally waterproofed (Kryton KIM) concrete slab roof that...
  48. H

    Finding the Dimension of a Vector Space: The Matrix Method

    Homework Statement A. Let {t,u,v,w} be a basis for a vector space V. Find dim(U) where U = span{t+2u+v+w, t+3u+v+2w, 3t+4u+2v, 3t+5u+2v+w} B. Compute the dimension of the vector subspace V= span{(-1,2,3,0),(5,4,3,0),(3,1,1,0)} of R^4Homework EquationsThe Attempt at a Solution I know that...
  49. M

    Random Walk in Arbitrary Dimension

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  50. amjad-sh

    Understanding Inner Product in Infinite Dimensional Bases

    While I'm reading a book in quantum mechanics, I reached the part "Generalization to infinite dimension". We know that at infinite dimension many definitions changes.And that what is confusing me! Take for example the inner product.when we are dealing in finite dimension the definition of inner...
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