What is Dimension: Definition and 897 Discussions

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. quasar987

    Compactness of the unit ball in infinite dimension

    Is it a fact that in an infinite dimensional normed linear space, the closed unit ball is never compact? If so, how does one go about seeing this?
  2. A

    The dimension of the span of three linearly independent R^3 vectors

    Homework Statement Given x_{1}= \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}, x_{2}= \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}, x_{3}= \begin{pmatrix} 2 \\ 6 \\ 4 \end{pmatrix} Find the dimension of Span(x_{1}, x_{2}, x_{3}) Homework Equations If V is a vector space...
  3. B

    Explaining the Fourth Dimension for Beginners

    Okay how can someone explain the four dimension in so that someone like me can under stand them? I'm asking this question because my math teacher will not talk about it so that the other students would not get confused. I get the theory (or how ever you want to say it)a little bit but i still...
  4. M

    The 4th Dimension: Exploring Human Curiosity & Its Theories

    Hello, I am in the process of reading Black Holes, Wormholes & Time Machines and am somewhat confused, even though it is aimed at people like myself who have very little knowledge on the subject. I am reading about the 4th dimension now but I do not understand the relevance of researching into...
  5. S

    Linear Algebra - Dimension of Kernel

    Homework Statement Suppose that U and V are finite-dimensional vector spaces and that S is in L(V, W), T is in L(U, V). Prove that dim[Ker(ST)] <= dim[Ker(S)] + dim[Ker(T)] Homework Equations (*) dim[Ker(S)] = dim(U) - dim[Im(T)] (**) dim[Ker(T)] = dim(V) - dim[Im(S)] The Attempt at a...
  6. M

    Dimension of a topological space

    In Hartshorne's book definiton of a dimension is given as follows: İf X is a t.s. , dim(X) is the supremum of the integers n s.t. there exist a chain Z_0 \subsetneq Z_1...\subsetneq Z_n of distinct irreducible closed subsets of X My question is: Can we conclude directly that any...
  7. T

    What is dimension of target space?

    what is dimension of target space?? is it the R^4 on top(which says that we have 4 coordinates like x y z t or 4 vectors in the matrix (whats it called the total number of the vectors??)
  8. F

    Proof of extra dimension from primordial BHs

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  9. G

    Exploring the 4th Dimension: My Theory on Energy's Role

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  10. W

    Motion of a Food Pack on a Rope

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  11. M

    Could time be movement in a 4th physical dimension?

    Let's say the universe exists as a collection of related static 3-d spaces, like in Julian Barbour's book The End of Time. Nothing ever actually moves. The illusion of movement comes from moving from one of these spaces to another which is closely related to the previous one. It's similar to the...
  12. K

    How far will a car coast up a slope before rolling back down?

    A car traveling at 16.0 m/s runs out of gas while traveling up a 21.0 degree slope. How far up the hill will it coast before starting to roll back down? I tried 1/2mv^2=mgh 1/2(16)^2=9.8h 13.06=h can anyone confirm if this is correct or not. I;m not sure if I have to use trig to...
  13. A

    Complete Dimension of Bevel Gear

    Homework Statement Hai Friends, i want to know the complete the dimension of bevel gear front teeth width and end teeth width. How this dimension are related with emperical formulas please give me complete detailing of bevel gear far straight and spiral also...
  14. N

    Dimension of column- and rowspace

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  15. A

    Equation Dimension Help: Valid Statements for Dimensionally Correct Equations

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  16. B

    From one space dimension to two space dimensions in special relativity

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  17. E

    Edward Witten on LQG for lambda > 0, QG does not exist in any dimension

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  18. E

    Dimension of the solution space for Ax=0

    Find the dimension of the solution space of Ax=0, where A=1 2 5 -1 3 1 is the rank(A)=2 so the nullity(A)=2? is this correct?
  19. P

    What is the dimension of a topological space?

    What is the exact definition of the dimension of a topological space?
  20. K

    The dimension of the fine structure constant

    Everyone "knows" that the fine structure constant (alpha) is a dimensionless number, but I am troubled by the fact that the Planck unit of resistance must be one - if that is true - and that strikes me as suspicious. I wonder if anyone can shed any light on this. I present the complete...
  21. J

    Pulley with Kinematics in One Dimension Question

    Homework Statement Two masses hanging vertically from a pulley are each initially 1.6 m above the ground and the massless frictionless pulley is 4.8 m above the ground. What maximum height does the lighter object reach after the system is released? Homework Equations \vec{F}_{net} =...
  22. I

    Homework question- kinematics in two dimension

    Ok I am stuck on how to solve this problem and i don't have physics again until next week. The question is: A ball is thrown horizontally at 22m/s from the roof of a building lands 36.0m from he base of the building. How tall is the building?
  23. E

    What is the dimension of the solution space for Ax=0 with A as a 2x3 matrix?

    Find the dimensions of the soultion space of Ax=0, where A= 1 2 5 –1 3 1 (A is a 2x3 matrix). To find the dimension you have to subtract n from rank(A) //n being the number of columns 3-2=1 The dimension of the solution space is 1. is this correct?
  24. N

    Correlation Dimension, what does it mean?

    I wasn't sure where precisely to put this, hopefully it gets an answer here. In my computational physics class we just learned about the correlation dimension. I know how to compute it, but don't quite know what it means. I learned roughly that the correlation dimension of something is a...
  25. Q

    Dimension of symmetric and skew symmetric bilinear forms

    Given the vector space consisting of all bilinear forms of a vector space V (let's call it B) it's very easy to prove that B is the direct sum of two subspaces, the subspace of symmetric and the subspace of skew symmetric bilinear forms. How would one go about determining the dimension of these...
  26. S

    Why our time dimension is about to become space-like

    It don’t get much weirder than this. The universe is about to lose its dimension of time says a group of theoretical astrobods at the University of Salamanca in Spain. And they got the evidence to prove it. [..] This ain’t entirely bonkers and here’s why. Bars Mars has calculated what it’s...
  27. A

    Find Physical Dimension of Constant C2

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  28. D

    Fortran Fortran: Passing integers to type dimension

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  29. J

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    Homework Statement A quarterback claims that he can throw the football a horizontal distance of 184 m. Furthermore, he claims that he can do this by launching the ball at the relatively low angle of 29.7 ° above the horizontal. To evaluate this claim, determine the speed with which this...
  30. F

    Dimension & Linear Maps: Does U=V?

    Hey guys, Does dimension remain unchanged under a linear map? Ie if i have a map f:U->V does dim(U) = dim(img(f))? Cheers -Z
  31. T

    How Does Projectile Motion Calculate Distance and Height?

    -------------------------------------------------------------------------------- A ball is tossed from an upper-story window of a building. The ball is given an initial velocity of 8m/s at an angle of 20 below the horizontal. It strikes the ground 3.00s later. The horizontal distance at which...
  32. T

    Calculating Distance and Height of a Tossed Ball in 2D Motion

    A ball is tossed from an upper-story window of a building. The ball is given an initial velocity of 8m/s at an angle of 20 below the horizontal. It strikes the ground 3.00s later. The horizontal distance at which the balls strike the ground from teh base of teh building and the height from...
  33. R

    3 dimension coordinate systems

    Homework Statement Describe in words the region of R3 (3 dimension) represented by the following inequality. 1)xyz=0 Homework Equations none i know of The Attempt at a Solution no idea where to start. I know that this means one variable must be equal to 0, but i don't...
  34. W

    How Do Runners, Rockets, and Jumpers Illustrate Principles of Motion?

    Homework Statement #1 Runner A is initially 4.0 mi west of a flagpole and is running with a constant velocity of 6.0 mi/h due to east. Runner B is initially 3.0 mi east of the flagpole and is running with a constant velocity of 5.0 mi/h due west. How far are the runners from the flagpole...
  35. Saladsamurai

    FINALLY A 4th dimension analogy for the layperson

    ...may frighten small children http://youtube.com/watch?v=6T0UQfKTcQw
  36. B

    Dimension magnetism problem

    In magnetism , mH = F, where m = pole strength ; H= magnetic field intensity ; and F = Force. F = [ MLT^{}-2] Proof that dimension of mH = [MLT^{}-2]
  37. U

    What is the dimension of this special stuff

    Homework Statement I am learning linear algebra (basic) and using Lang's book. In talking about vector spaces (finite dimensional only), it is easy to understand what is the dimension of the vector space (i.e. the number of elements in any basis). But I met the following case (not a vector...
  38. J

    What is the Meaning of Dimension

    What is the meaning of dimension? It is my understanding that dimension related to the measuring out of something. If there is some observable phenomena that we can measure by defining units of measure and counting the quantity of these units that there is an associated dimension which is not...
  39. R

    New Dimension: Exploring a Black Hole Passage

    A new DIMENSION?? To Support the Theory of going into another dimension by traveling through a Black Hole. Since Scientists say you can go to another dimension through a Black Hole, why not you go to another dimension by just going pass a Black Hole?:confused:
  40. E

    Simple Problem in One Dimension

    Homework Statement Exercise 5.1.1 : Show that Eq (5.1.9( may be rewritten as an integral over E and a sum over the +/- index as U(t) = \sum_{a=\pm}\int^{\infty}_{0}|E,a><E,a|e^{-iEt/\hbar}dE You will find at...
  41. S

    Imagining the tenth dimension

    Hey a friend linked me to this video: http://www.youtube.com/watch?v=2u4CFTZtcXE" which attempts to explain how the idea of a tenth dimension is feasible. I have had some trouble understanding some of the concepts which the movie attempts to explain and it made me think of many questions...
  42. J

    Time as a Dimension: Is Time Expanding?

    This may be a dumb question, but if the universe is expanding and time can be treated as just another dimension, does this mean that time is 'expanding'? And if so, what the heck does that mean?
  43. D

    Fortran How many bytes in DIMENSION? - A Fortran Question

    How many bytes in DIMENSION?? - A Fortran Question I'm looking at some Fortran code. Some variables are of type dimension. How do I tell if these variables are integers or floating points numbers? How do I know if they are signed or unsigned? How do I tell the number of bytes occupied by...
  44. J

    Vector space dimension of little-l 2

    Homework Statement I'm trying to understand why \ell_2^\infty as a vector space over \mathbb{C}, has uncountable dimension. Homework Equations The Attempt at a Solution Firstly, I'm not really clear on the meaning of basis in infinite dimensions. Is it still true that any element...
  45. L

    Football player kinematics in one dimension

    Homework Statement A football player, starting from rest at the line of scrimmage, accelerates along a straight line for a time of 3 seconds. Then during a negligible amount of time, he changes the magnitude of his acceleration to a value of 1.1 m/secondssquared. With this acceleration, he...
  46. M

    Kinematics in two Dimension- projectile motion

    I am really having trouble solving this problem. Can someone help ASAP?? An airplane with a speed of 87.9 m/s is climbing upward at an angle of 59.8 ° with respect to the horizontal. When the plane's altitude is 652 m, the pilot releases a package. (a) Calculate the distance along the ground...
  47. B

    Two dimension relative morion help

    Two dimension relative morion help! Homework Statement After flying for 15 min in a wind blowing 42 km/h at an angle of 18° south of east, an airplane pilot is over a town that is 60 km due north of the starting point. What is the speed of the airplane relative to the air? Homework...
  48. Robert100

    How is 3-space curved without a fourth spatial dimension?

    When I was studying physics in college I learned two contradictory ways to understand Einstein's theory of general relativity (GR). The most common way was to visualize our three-dimensional space as being distorted by mass. Anything with mass created a distortion in spacetime. Just as a...
  49. K

    Rank, Dimension, Subsapce, Column Space

    1) True or False? If true, prove it. If false, prove that it is false or give a counterexample. 1a) If A is m x n, then A and (A^T)(A) have the same rank. 1b) Let A be m x n and X E R^n. If X E null [(A^T)(A)], then AX is in both col(A) and null(A^T). [I believe it's true that AX is in...
  50. G

    Exploring the 6th Dimension: A 10th Grader's Perspective

    Ok, so I'm new to this message board and I'm only in 10th grade so please bere with me. So being from a small town I've decided to take on a big project and write a research paper on the Sixth Dimension. Here is my understanding of the dimensions so far: Zeroth Dimension= a point, consisting of...
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