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. C

    Comparing Velocities and Accelerations of Two Motorcycles in One Dimension

    Homework Statement Two motorcycles are traveling due east with different velocities. Four seconds later, they have the same velocity. During this 4s interval, motorcycle 1 has an average acceleration of 2.0 m/s(squared) due east, while motorcycle 2 has an average acceleration of 4.0...
  2. L

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    Homework Statement A Pendulum weighs .5kg and has a string length of 70cm swings from a horizontal position downwards to hit a block that weighs 2.5 kg and is on a frictionless plain. Calculate the speed of both the ball and the block after the elastic collision. Homework Equations...
  3. M

    Proving W=V When W is a Subspace of an n-Dimensional Vector Space

    Homework Statement Prove that if W is a subspace of an n-dimensional vector space V and dim(W) = n, then W=V Homework Equations The Attempt at a Solution I don't know where to start.
  4. J

    Linear Algebra: dimension of subspace question

    Homework Statement Find an example of subspaces W1 and W2 in R^3 with dimensions m and n, where m>n>0, such that dim(intersection of W1 and W2)= n Homework Equations dim(W1+W2)= dim(W1) + dim(W2)-dim(intersection of W1 and W2) The Attempt at a Solution Well what I know...
  5. Y

    Acceleration in one dimension

    Homework Statement "2 cars C and D travel in the same direction in straight line. During particular time interval t, car D is AHEAD of car C, and D is speeding up while car C is slowing down. During the interval t, it is observed that the distance between the cars decreases. Explain how this...
  6. H

    Dimension of Angle: Learn LL^-1 from Course Book

    In my course I am curios to know how the dimension of angle becomes [LL^-1]. The following pic is taken by course book. http://i42.tinypic.com/iy2y3r.jpg
  7. K

    Time as Dimension: Before & After Einstein's G.T.R.

    Since when is Time considered to be a dimension? Is it after Einstein's G.T.R. or was it considered to be one before it?
  8. J

    What is the dimension of its kernel?

    Homework Statement Suppose that dim V = m and dim W = n with M>=n . If the linear map A : V -> W is onto, what is the dimension of its kernel? Homework Equations The Attempt at a Solution Onto, means that every vector in W has at least one pre-image therefore, the kernel can...
  9. B

    Subspace & Dimension of V in P2 - Closure and Dimension Analysis

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  10. M

    How Long Does It Take for Two Rocks Thrown from a Bridge to Hit the Water?

    Homework Statement Heather and Jerry are standing on a bridge 46 m above a river. Heather throws a rock straight down with a speed of 30 m/s. Jerry, at exactly the same instant of time, throws a rock straight up with the same speed. Ignore air resistance. How much time elapses between the...
  11. M

    Kinematics in One Dimension

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  12. marcus

    Nice LQG paper by Modesto about dimension down with scale

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  13. S

    Electromagnetic force in the fourth spatial dimension?

    Purely from a non-mathematical perspective, could some/any/every-body answer my questions on the electromagnetic force in extra dimensions? άλφα) I am curious about how in the old Kaluza theory of the electromagnetic force the equation, which he found from his equations worked when it...
  14. H

    How long did it take for the box to travel 4.00 meters?

    Homework Statement A box slides along a surface with a positive initial velocity. It the experiences an acceleration of -0.25m/s^2. After traveling 4.00 meters, its velocity is +0.50m/s. How long did it take for the boxto travel the 4.00 meters? Homework Equations \Deltax=Vot + 1/2 a t^2...
  15. Spinnor

    Suppose there is an extra compact dimension, and suppose

    Suppose there is an extra compact spatial dimension in addition to familiar space dimensions x, y, and z. Let us suppose that matter is some kind of 3 dimensional surface moving in these 4 spatial dimensions, 4-space, in some cyclical manner. Let us suppose that from the shape of the surface we...
  16. J

    Time as a Uni-Dimensional Force

    Hi there, I have a question. To me, measuring spatial dimensions is quite a bit different than measuring the time dimension. I was thinking the other day and it seemed very obvious to me what the main difference was. Time never goes the other way. A dimension defines two equal and...
  17. P

    Linear Algebra: Null Space and Dimension

    Homework Statement Prove that dim(nullA) = dim(null(AV)) (A is a m x n matrix, V is a n x n matrix and is invertible Homework Equations AX=0 and AVX = 0 Null(AV) = span{X1,..Xd} Null(A) = span{V-1X1,.., V-1Xd} The Attempt at a Solution so you need to prove that...
  18. Z

    Could GR generalized to non-integer dimension?

    could GR generalized to non-integer dimension?? let us suppose that the dimension of space time is NOT an integer then , could we generalize GR to obtain an expressions of Tensor, Covariant derivatives... in arbitrary dimensions ?? let us say 4.567898.. or similar, i mean GR in non integer...
  19. O

    Range of the Hausdorff dimension

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  20. D

    Optimization - Find dimension of a cup that uses least amount of paper

    Homework Statement A cone-shaped paper drinking cup is to be made to hold 30 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper. Homework Equations volume of a cone (1/3)(pi)(r^2)(h) = 30 SA of a cone pi(r)[sqrt(r^2 + h^2)] The Attempt at...
  21. R

    Dimension and Orthogonality in Vector Spaces: A Proof of the Inequality m ≤ n

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  22. T

    Reynolds number - Characteristic dimension

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  23. U

    Dimension of preimage question

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  24. A

    Vector Space Dimension: Real vs Complex Coefficients

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  25. T

    Reducing the pipe dimension to a heating device

    Let's say there is a water based heating device that has 3/4" connection nipples. The pipe that leads to the device is 5/4". If one ignores pressure- and head loss, does it make any difference how far from the device the pipe reduces from 5/4" to 3/4" ?
  26. A

    What is the null space basis and dimension of A in R^5?

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  27. K

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  28. Mentallic

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  29. E

    Divergence theorem in four(or more) dimension

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  30. C

    Dimensions Creation: Theory or Always Existed?

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  31. E

    Collision in two dimension

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  32. B

    Volume of Dimension reduced due to gravitation

    I've had this on my mind for a while. I'm pretty confident that I have the right idea, but I can't find a formula that I could use to prove it to someone. So, here it is. Imagine a sphere of space-time with absolutely nothing inside of it. It is a large sphere, sun-size large. Now, take...
  33. B

    Is there a formula for the shrinking of the universe due to object curvature?

    Earlier, I had a thought, and I have now been curious about the equation of this thought. Einstein's general relativity obviously states that objects curve and distort space. But the whole reason he came up with a cosmological constant was that he found that the curvature of all of the objects...
  34. C

    Dimension Proof: U + W = dim(U) + dim(W) - dim(U ∩ W)

    Homework Statement Let U, W be subspaces of a vector space V. Show that dim(U+W) = dim(U) + dim (W) - dim (U intersect W). Homework Equations The Attempt at a Solution I can see this picture-wise in a venn-diagram form. In adding U and W you count the elements in their...
  35. Y

    Answer: Lie Algebras: Adjoint Representations of Same Dimension as Basis

    Hello, I hope it's not the wrong forum for my question which is the following: Is there some list of Lie algebras, whose adjoint representations have the same dimension as their basic representation (like, e.g., this is the case for so(3))? How can one find such Lie algebras? Could you...
  36. Ivan Seeking

    HG Wells: The first to call time a fourth dimension?

    The claim was made tonight on a program about science fiction writers that HG Wells was aware of a geometry of 4 dimensions, and then made the leap of faith that time was a 4th dimension - ten years before Einstein came along. Did he really conceive of this before anyone else? I never read...
  37. D

    Solving Dimension Analysis Homework: Q1 & Q2

    Homework Statement http://img410.imageshack.us/img410/8495/88748860vk8.png Homework Equations The Attempt at a Solution I don't know how to deal with the q1 part. q2 = u* =u x sqrt(... so that is clear. Normally I just substitute the expression for u in terms of u* (= f* )and t in terms of...
  38. D

    Dimension Analysis Homework: Buckingham Theorem

    Homework Statement http://img21.imageshack.us/img21/613/70858934fn5.png Homework Equations Buckingham theoremThe Attempt at a Solution My question is what is the difference between question a and b? The sine doesn't influence the dimension. Or is it a question to trick me?
  39. G

    Dimension of U,V and W over K: Do they Equal?

    Just have a question if U,V and W are over the same field K does it mean that dim U = dim V = dim W ?
  40. P

    Understanding Matrix Dimensions: Calculating Entries and Recognizing Symmetry

    How do we calculate the dimension of a matrix? Is it the number of entries? Or is it the number of different entries? For instance if I have a matrix 2x2 the dimension would be 4 but if the matrix is simetrical it would be 3. Is this correct? Thanks for your help.
  41. J

    Dimension of row/ column space

    Homework Statement In the following exercises verify that the row rank is equal to the column rank by explicitly finding the dimensions of the row space and the column space of the given matrix. A = [1 2 1 ; 2 1 -1] Homework Equations The Attempt at a Solution All i can...
  42. J

    Motion in One Dimension - Finding two separate changes in time

    Homework Statement A commuter train travels between two downtown stations. Because the stations are only 1.30 km apart, the train never reaches its maximum possible cruising speed. During rush hour the engineer minimizes the travel interval Δt between the two stations by accelerating for a...
  43. A

    Calculating Overtaking Time and Distance in One-Dimensional Motion

    Kathy Kool buys a sport car that can accelerate at the rate of 4.90 m/s2. She decides to test the car by racing with another speedster, Stan Speedy. Both start from rest, but experienced Stn leaves the starting line 1.00 s before Kathy. Stan moves with a constant acceleration of 3.50 m/s2 and...
  44. M

    Kinematics 1 Dimension, Person Running Solve for Time

    Homework Statement A runner hopes to compleate the 10,000m run in less than 30.0min. After running at a constant speed for exactly 27.0min there are still 1100m to go. The runner must then accelerate at 0.20 m/s^2 for how many seconds in order to achieve the desired time. *note* person does...
  45. D

    Is Gravity Considered a Dimension in Physics?

    Hey everyone. I was recently on youtube, looking at 4-d objects, when I came across a comment which caught my eye: "You're forgetting Gravity. Without gravity there is no xyz and without two large bodies there is no concept of time. Therefore, gravity must be included as a dimension. They are...
  46. S

    Why phase transition not possible in one dimension?

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  47. E

    Runner's motion in one dimension

    Homework Statement A runner hopes to complete the 10,000-m run in less than 30.0 min. After exactly 27.0 min, there are still 1100 m to go. The runner must then accelerate at 0.20 m/s2 for how many seconds in order to achieve the desired time? Homework Equations vf = vo + at avg...
  48. B

    An equation related to the dimension of linear operator

    Homework Statement Let V be a finite vector space, and A, B be any two linear operator. Prove that, rank A = rank B + dim(Im A \cap Ker B) The Attempt at a Solution Since rank A = dim I am A dim(Im B)+ dim(Ker B)=dim V dim(Im A + Ker B)=dim(Im A)+dim(Ker B)-dim(Im A \cap Ker B) It...
  49. C

    Time - a Dimension of Physics or Mathematics?

    I have read a number of threads that argue the ontological status of time in physics. One of the most concrete "descriptions" of time in physics is offered by Huw Price. Huw Price suggests time is symmetrical, but the order of events in time are not. http://arxiv.org/abs/physics/0402040...
  50. I

    Trace(matrix) = 0 and the dimension of subspace

    Homework Statement Claim: Matrices of trace zero for a subspace of M_n (F) of dimension n^2 -1 where M_n (F) is the set of all nxn matrices over some field F. Homework Equations Tr(M_n) = sum of diagonal elements The Attempt at a Solution I view the trace Tr as a linear...
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