What is Dimension: Definition and 899 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. D

    Dimension of subspace

    C[-pi, pi] Span(1, cos (2x), cos2 (x)) Doing the Wronskian here is pain so what other method would be more appropriate?
  2. S

    Dimension of intersection of U and V

    Homework Statement Prove true: For any subspaces U,V of R^n dim(U intersect V) <= min(dim(U), dim(V)) Homework Equations Min(a,b) = the minimum value of A and B The Attempt at a Solution I know this statement is true however I can't quite figure out where to start on how to...
  3. M

    Dimension of Hom(K)(U,V) and Basis of the Vector Space

    Homework Statement Let U and V be vector spaces of dimensions of n and m over K and let Hom(subscriptK)(U,V) be the vector space over K of all linear maps from U to V. Find the dimension and describe a basis of Hom(subscriptK)(U,V). (You may find it helpful to use the correspondence with mxn...
  4. E

    Finding the dimension (Linear Algebra)

    Homework Statement My question is how to find the dimension of something. Let's say: x1-2x2+x3-x4=0 and let's do one with more than 1 equation. x1+x3-2x4=0 x2+2x3-3x4=0 Homework Equations The Attempt at a Solution For the second one I figured that if I took the...
  5. O

    How to determine the number of supercharges (N) for any dimension (D)?

    e.g. For D=11, the M-theory, we require N=8, ... For D=10, we require N=16, or 32... ... So how to determine the number of supercharges, the \thetas, for arbitrary dimension, e.g. D=1, 2, ...
  6. Y

    Question regarding D'Alembert solution for one dimension wave equation

    In am studying PDE and I have question about D'Alembert solution for one dimension wave equation. I am going to reference Wolfram: http://mathworld.wolfram.com/dAlembertsSolution.html 1) I want to verify the step of \frac{\partial y_0}{\partial t} of step (14) of the page...
  7. M

    Basis and Dimension of Subspace V

    Homework Statement V = the set of all symetrical nXn matrices, A=(ajk) such that ajk=akj for all j,k=1,...,n Determine the base and dimensions for V The Attempt at a Solution I set my matrix up as [a11 a12] [a21 a22] So a21 and a12 are equal to each other? I assume the...
  8. I

    What Is the Dimension of Subspaces U and W in a Vector Space V?

    Homework Statement V=R^{4}\ and\ a^{\rightarrow}, b^{\rightarrow}, c^{\rightarrow}, d^{\rightarrow}, e^{\rightarrow} \in V. (I'll drop the vector signs for easier typing...) a = (2,0,3,0), b = (2,1,0,0), c = (-2,0,3,0), d = (1,1,-2,-2), e = (3,1,-5,-2) Let\ U \subseteq V be\...
  9. M

    Solving for Vector Space V: Find Dimension & Basis

    Homework Statement Find the dimnesion and a basis of vector space V Homework Equations V is the set of all vectors (a,b,c) in R^3 with a+2b-4c=0 The Attempt at a Solution (4c-2b,b,c) = b(-2,1,0) + c(4,0,1) so {(-2,1,0),(4,0,1)} is the basis of the SUBSPACE of V right? how do I...
  10. P

    How Can a Train Travel 1km Optimally by Accelerating and Braking?

    Homework Statement A commuter travels between two stations. the stations are only 1.00km apart which means the train never reaches maximum crusing speed. The engineer minimizes the time interval (delta t) between the two stations by accelerating for a time interval (delta t sub 1) at a rate...
  11. C

    Dimension and basis for subspace determined by given vectors

    Homework Statement Assume that e_1 ,..., e_n is a basis for the vector space V. Let W be the linear subspace determined (formed?) by the vectors e_{1}-e_{2}, e_{2}-e_{3}, ..., e_{n-1}-e_{n}, e_{n}-e_{1}. Determine the dimension of W, and a basis for W. Homework Equations The...
  12. D

    What is the dimension of c3 in s=cs cos(c4t)

    Find the dimension for the quantity c3 in the expression s=c3 cos (c4t). Please someone help me solve this, s is a distance with unit L, t is a time with unit T and theta is an angle in radians.
  13. T

    How to find the domain in one dimension

    Homework Statement I am trying to find the domain of the following.. R^2 to R^3 of g(x,y) = (x-y,x+y,3*x) R^3 to R of h(x,y,z) = x/(y+z) Homework Equations The Attempt at a Solution I don't know how to start. I know how to find the domain in one dimension but how...
  14. M

    Nature of the time dimension in 4D space-time

    I read on a website (http://math.ucr.edu/home/baez/physics/Relativity/GR/gravity.html): "The world we live in consists of four dimensions, the three space dimensions and one that is not exactly time but is related to time (it is in fact time multiplied by the square root of -1)." In my...
  15. A

    Information is a kind of dimension (like space and time)

    information is a kind of "dimension" (like space and time) I always asked myself about how does information works in the cosmos. I read an article were some scientist told that information is a kind of "dimension" (like space and time), and its crucial for the conformation of systems (and...
  16. E

    What Defines a Dimension? Exploring Length, Width, and Depth in Space

    Hey everyone, I'm not a physicist, I'm a high school senior with a who stopped taking math and science after sophomore year (which I sorely regret). So, I have a question rather than a crazy hypothesis regarding the primordial cocktail antecedent to the big bang that's dubious at best. Call...
  17. R

    2 dimensional PDE and a higher dimension problem.

    Homework Statement The 2d PDE Assume f\in S(\mathbb{R}^2) (Schwartz space) Then solve u_{xx}(x,y) + 2u_{yy}(x,y) + 3u_{x}(x,y) -4u(x,y) = f(x,y) ; (x,y) \in\mathbb{R}^2 u_{xxxx}(x,y) - u_{yy}(x,y) + 2u(x,y) = f(x,y) ; (x,y) \in\mathbb{R}^2 Homework Equations The relevant...
  18. R

    Dimension of subspace of V^n with orthogonal vectors

    in a space V^n, prove that the set of all vectors {v1,v2,..}, orthogonal to any v≠0, form a subspace V^(n-1). i know that a subspace of V^n must be at least one dimension less and the set of vector v1,v2,... build a orthogonal basis, but how can one show with this preconditions that the...
  19. B

    Sard theorem in dimension one.

    Hi, I need some hints for the proof of the sard theorem in 1 dimension: Prove that the set of critical values (f(x) where f'(x)=0) of continuously differentiable f:[a,b]->R has measure 0.My attempt:Fix \varepsilon, Let Crit(f) be the set of critical points of f. We want to show that...
  20. B

    Proof dim(U+V)=dim U+dim V - dim(U∩V)

    let U and V be subspaces of Rn. Prove that dim(U+V)=dim U+dim V - dim(U∩V)
  21. I

    Dimension of the gamma matrices in general dimension

    Dear guys, I read a derivation of the dimension of gamma matrices in a d dimension space, which I don't quite understand. First of all, in d dimension, where d is even. One assumes the dimension of gamma matrices which satisfy \{ \gamma^\mu , \gamma^\nu \} = 2\eta^{\mu\nu}...
  22. J

    Difference between dimension and rank

    Hey all. I know this is a basic concept but I don't really understand it. I don't get what the difference between rank and dimension is. According to my book, the rank of a matrix is the dimension of the column space. Does that not imply that they are the same, unless the question...
  23. Z

    Dimension of a subspace of polynomials with certain coefficients

    Right so I've had an argument with a lecturer regarding the following: Suppose you consider P_4 (polynomials of degree at most 4): A(t)=a_0+a_1t+a_2t^2+a_3t^3+a_4t^4 Now if we consider the subspace of these polynomials such that a_0=0,\ a_1=0,\ a_2=0}, I propose that the dimension of of this...
  24. A

    Relativity and Spatial Dimension

    If mass influences the 'shape' of three dimensional space plus time, does that require a further or higher spatial dimension, or is that unnecessary?
  25. B

    D dimension scalar potential for point charge

    Homework Statement Show that with d spatial dimensions the potential \phi due to a point charge q is given by \phi (r) = \frac{\Gamma(\frac{d}{2}-1)}{4\pi^{d/2}}\frac{q}{r^{d-2}} Homework Equations The Attempt at a Solution The electric field strength is known to be: E(r) =...
  26. B

    Compute transformation matrix in nth dimension

    I have a vector of all ones in n-dimensions. For example (1,1,1) in 3D. I want to find a invertible rotation matrix T that transforms the vector of all ones to the vector (0,0,0,...,0,,1): Let v be the vector of all ones, and w=(0,0,...,0,1) Find T such that T.v == wIn low dimension it is easy...
  27. P

    Consciousness and time dimension

    IF consciousness is a result of complex physical processes, and could thus be simulated by a hugely complex computer program (which should not be extremely controversial) - WHAT IF the finite execution of the computer program would be presented, not with temporal steps (using the time...
  28. C

    A problem with basis and dimension

    Hi! I am working on the following problem: If a matrix is antisymmetric (thus A^T = -A), show that P = {A \in R | A is antisymmetric} is a subset of Rnxn. Also, find the dimension of P. So far, I've proven that P is a subset of R and I am guessing that, in order for me to find the...
  29. S

    Linear algebra. Problem with vector spaces dimension

    Homework Statement First of all sorry if my terminology sounds a bit weird, i have never studied mathematics in english before. So this is the problem: We have the space R^2x2 of all the tables with numbers in R. We also have a subspace V of R^2x2 of all the tables with the following...
  30. M

    Exploring Time as a Testable Dimension

    Just a quick question. I know that it is pretty much taken to be axiomatic, but is time a testable dimension?
  31. S

    Harmonic motion in 1 dimension

    Homework Statement 1. A particle of mass m is constrained to move along a straight line. In a certain region of motion near x = 0 , the force acting on the particle is F = -F_0 sin(bx) , where F_0 and b are positive constants. (a) Find the potential energy of the particle in this region...
  32. P

    Elastic Collisions in one dimension

    Homework Statement Particle 1 of mass m1 = 0.30 kg slides rightward along an x-axis on a frictionless floor with a speed of 2.0 m/s. When it reaches x = 0, it undergoes a one-dimensional elastic collision with stationary particle 2 of mass m2 = 0.40 kg. When particle 2 then reaches a wall...
  33. M

    Elastic collision in one dimension

    Homework Statement Block 1 of mass m1 slides along an x-axis on a frictionless floor with a speed of 2.40 m/s. Then it undergoes a one-dimensional elastic collision with stationary block 2 of mass m2 = 2.00m1. Next, block 2 undergoes a one-dimensional elastic collision with stationary block...
  34. K

    What dimension is the surface of a cylindrical shell

    Hello there, i wanted to know what is the difference between a surface in R^2 and a surface in R^3 in an xyz cordiante system. More specifically how do they look like? an example please. Is this different from a 2 dimensional surface and a 3 dimensional surface respectively. Also what...
  35. T

    Exploring the Concept of Time as a Dimension in Physics

    I've often heard that time was considered a dimension, but I don't understand how this is possible as it invalidates the idea of motion. Take a simple number line for example. This is considered the first dimension. A point can be at any place on said number line. Moving on up to the second...
  36. G

    Elastic Collisions in One Dimension

    Homework Statement A cart with mass 310 g moving on a frictionless linear air track at an initial speed of 1.4 m/s undergoes an elastic collision with an initially stationary cart of unknown mass. After the collision, the first cart continues in its original direction at 0.64 m/s...
  37. S

    Potention in one dimension Help

    Can anyone help me with a problem. Please just answer whatever you can. Thanks. I have not started the problem because I don't know where to begin. I can solve physics problems but i just can't seem to start any of them off. A classical particle of mass m moves in the presence of the...
  38. B

    Linear Algebra: Basis and Dimension

    Homework Statement In each case, check that (v1,...vk) is a basis for Rn and give the coordinates of the given vector b belongs to Rn with respect to that basis. Homework Equations a) v1=(2,3) v2=(3,5) b=(3,4) b) v1=(1,0,1) v2=(1,1,2) v3=(1,1,1) b=(3,0,1) The Attempt at a Solution I...
  39. Z

    Kinematics in One Dimension test

    This is not a homework problem, I am not sure what section to put this (I apologize if it's in the wrong place). So I take AP Physics B at my high school and we just got our first test back on Kinematics in One Dimension and I got a 53%! I know that might sound horrific (which it is) but my...
  40. P

    How Deep Is the Well If a Rock Takes 2.4 Seconds to Hit the Water?

    A rock is dropped from rest into a well. The sound of the splash is heard 2.40 s after the rock is released from rest. a)How far below the top of the well is the surface of the water? The speed of sound in air (at the ambient temperature) is 336 m/s. (b) What If? If the travel time for...
  41. K

    How Far Apart Are the Stop Signs?

    Homework Statement A car starts from rest at a stop sign. It accelerates at 4.0 m/s^2 for 6.0 s, coasts for 1.9 s, and then slows down at a rate of 3.0 m/s^2 for the next stop sign. How far apart are the stop signs? Homework Equations The Attempt at a Solution
  42. G

    Solving Schrodinger Equation in Two Dimension, r and theta

    Homework Statement It's a two-part problem, the first part was deriving a Schrodinger equation from when x = r cos(theta) and y = r sin(theta) I got: -\frac{\hbar^2}{2m}[\frac{\partial^2}{\partial...
  43. S

    Calculating Time for an Object to Fall in One Dimension

    Hi. I am new to Physics Forums. The following problem is from Physics 6th Edition by Cutnell/Johnson. A hot air balloon is rising upward with a constant speed of 2.50 m/s. When the balloon is 3.00 m above the ground, the balloonist accidentally drops a compass over the side of the balloon...
  44. V

    Motion in one dimension physics problem

    Homework Statement Determine the stopping distances for an automobile with an initial speed of 90 and human reaction time of 1.0 : (a) for an acceleration = -5.0 , (b) for = -7.5 . Homework Equations vf = vo + at avg velocity = (vf + vo) /2 d = vo)t + (1/2) at2 vf2 = vo2 + 2ad...
  45. V

    Motion in one dimension problem

    Homework Statement An unmarked police car traveling a constant 85 is passed by a speeder. Precisely 1.00 after the speeder passes, the police officer steps on the accelerator. If the police car accelerates uniformly at 3.00 and overtakes the speeder after accelerating for 5.00 , what was the...
  46. N

    Constant Acceleration one dimension

    Homework Statement A rocket starting at rest takes on a net acceleration of 20m/s^2 in a vertical line until it runs out of fuel after 5 seconds. At what height does it run out of fuel? What is its velocity when it runs out of fuel? What is its maximum height? How long does it take to hit the...
  47. A

    How do I calculate the height of a tower using motion in one dimension?

    I have a few problems I'm having trouble with. If I can get some help with this one I should be able to figure out the rest I have. 1. A rock is thrown downward from the top of a tower with an initial speed of 12 m/s. If the rock hits the ground after 2.0 s, what is the height of the tower...
  48. L

    What is the fourth variable in the 4th dimension?

    is it time? what's your theory on the study of other dimensions? and applications from them?
  49. nomadreid

    Fractal dimension of CMBR? Cluster distribution?

    I have read speculations that (1) the cosmic microwave background radiation has a fractal distribution (non-integral Hausdorff dimension), and (2) the same might be true of galaxy cluster distribution (although different dimensionality to (1)) Whether or not one or both analyses are...
  50. J

    Primary Dimension: Mass Units for Power, Pressure, Modulus, Angular Velocity

    Homework Statement For each quantity listed, indicate dimensions using mass as a primary dimension and give typical SI and English units: power pressure modulus of elasticity angular velocity Homework Equations The Attempt at a Solution im not sure i understand what it is...
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