Dimension Definition and 908 Threads

From what I know, in 2 dimensions gravity works as 1 / R, 3 dimensions is 1 / R^2, and in 4 dimensions 1 / R^3 But what about one dimension? Is it just proportional to R? So the formula would be F = GMMR? Please explain in detail, how gravity would behave in one dimension and how...

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?

Hi everyone! I am thinking about, how can calculate the Hausdorff dimension of the Cantor set? I know, that this dimension is \frac{\log 2}{\log 3} but I cannot prove it. Any ideas?

Hi everyone, The problem that I'm having issues with reads: "Let F: Rk →Rn be a linear map. Recall that the graph G(F) of F is the subset of Rk × Rn = Rk+n given by G(F)={(x,y)∈Rk ×Rn : y=F((x)}" It first asked me to show that G(F) is a vector subspace of Rk+n which I did just by the definition...

Homework Statement Given is a vector space (V,+,k) over kn with n > 1. Show that with W \subseteq V, U \subset V and dim(U) = n - 1 dim(W \cap U) \geq dim(W) - 1 Homework Equations dim(W+U) +dim(W \cap U) = dim(W) +dim(V) The Attempt at a Solutiondim(V) = n dim(W) \leq dim(V) dim(W+U)...

Homework Statement I have a question which asks me to find the dimensions of the subspace of even polynomials (i.e. polynomials with even exponents) and odd polynomials. I know that dim of Pn (polynomials with n degrees) is n+1. But how do I find the dimensions of even n odd polynomials...

Let V=Mn(k) be a vector space of matrices with entries in k. For a matrix M denote the trace of M by tr(M). What is the dimension of the subspace of {M\inV: tr(M)=0} I know that I am supposed to use the rank-nullity theorem. However I'm not sure exactly how to use it. I know that the trace is...

Homework Statement Find the basis and dimension of the following subspace U of P2 p(x) \ni P2 such that p(1) = p(2)Homework Equations The Attempt at a Solution I know all quadratics are in the form ax2 + bx + c set p(2) = p(1) 4a + 2b + c = a + b + c b = -3a Therefore ax2 -3a + c Basis(U)...

Homework Statement Derive formula for elastic collisions in one dimension, unequal masses target at rest. i am having trouble acquiring the same formula that the book gives me, please tell me where my algebra is incorrect... i need to be able to derive these formulas properly because...

Homework Statement A car travels north at 30 m/s for one half hour. It then travels south at 40 m/s for 15 minutes. The total distance the car has traveled and its displacement are? Homework Equations The Attempt at a Solution First of all i converted from m/s to km/h. I took...

Hello, please any help will be very much appreciated; A bullet is fired through a board, 14.0 cm thick, with its line of motion perpendicular to the face of the board. If it enters with a speed of 450 m/s and emerges with a speed of 220 m/s, what is the bullet's acceleration as it passes...

Could someone explain the fourth dimension to me? I understand that it represents time. Could anyone thoroughly explain the concept? Thank you.

So a particle of mass M is moving in one dimension given by: x(t) = (alpha)t^2 + (beta)t + (gamma) where alpha, beta, gamma are non zero constants. What are the dimensions of the alpha, beta, gamma? Either the answer is staring right at my face or my physics is rusty :/

dimension of sl(2,R) = 1*(2*2-1) = 3, is isomorphic to so(2,1) : 2+1 = 3 dimension of sl(2,C) = 2*(2*2-1) = 6, is isomorphic to so(3,1) : 3+2+1 = 6 dimension of sl(2,H) = 15, is isomorphic to so(5,1) : 5+4+3+2+1 = 15 dimension of sl(2,O) = 45, is isomorphic to so(9,1) : 9+8+7+6+5+4+3+2+1 = 45...

This should be a simple question for anyone familiar with basic algebraic geometry, but the concept is getting the best of me. Basically I am unclear on the concept of dimension of a variety. I'm still stuck in the algebraic mindset that dimension is the (minimum) number of elements needed to...

Homework Statement In an action-adventure film, the hero is supposed to throw a grenade from his car, which is going 79.0 km/h , to his enemy's car, which is going 116 km/h . The enemy's car is 16.1m in front of the hero's when he let's go of the grenade. If the hero throws the grenade so...

I read Stephen Hawking new book and did a little research on M-theory. Apparently in the 11th dimension there are different brains that crash into each other. 2 of these brains crashing supposedly rendered our big bang, and explains why stuff came out in blobs (not homogeneous). But how...

Homework Statement A car accelerates at 2.10 m/s2 along a straight road. It passes two marks that are 29.6 m apart at times t=4.10 s and t=4.90 s. What was the car's velocity at t=0? I'm assuming the car has constant acceleration of 2.10 m/s2 Given \Deltax = 29.6 m a = 2.10 m/s2 vi = ...

Homework Statement A hot air balloon is ascending straight up at a constant speed of 8.10 m/s. When the balloon is 13.0 m above the ground, a gun fires a pellet straight up from ground level with an initial speed of 28.0 m/s. Along the paths of the balloon and the pellet, there are two...

i am now studying dirac equation and klein paradox if we confine to one dimension, we only need one alpha matrix, not three so in lower dimensions, maybe the dirac spinor is not of four components but fewer? i am curious about this question because it seems that as for the Klein...

So do we know for sure what the 4th spatial dimension is? I have read in many places that it is time. I would say this makes sense because we are perceiving time in 3 dimensional slices in our dimension. The same way a 2D being would perceive 2D slivers of a 3D object in its own dimension.

I see lots of references to time being the fourth dimension as well as there being 3 + 1 dimensions to spacetime as we know it, etc. I also see that time has to be treated differently in some of the constructs of physics. So it seems that time seems to be both similar and dissimilar to the other...

Suppose I have a basis for a subspace V in \mathbb{R}^{4}: \mathbf{v_{1}}=[1, 3, 5, 7]^{T} \mathbf{v_{2}}=[2, 4, 6, 8]^{T} \mathbf{v_{3}}=[3, 3, 4, 4]^{T} V has dimension 3, but is in \mathbb{R}^{4}. How would one switch basis for this subspace, when you can't use an invertible...

In Minkowski 4-D spacetime the cooridnates of an point are (ct,x,y,z). My question is why we multiply time with speed of light c and not some other speed v (speed of object moving in three spatial dimantions)? If we treat the time as a fourth dimension it would be more reasonable to say that...

Dear all, Is there a method to find the maximum and minimum dimension of an irregular closed loop? This is a problem when we want to define the full-width - half maximum of a image. The level contour of this image at its half maximum can be an irregular closed loop. Any reference or...

(Maybe this should go under General Math or maybe even Topology but since it's about the dimension of the universe I'll put it here. Feel free to move it.) I've had too much coffee and on one my frequent wiki binges - reading about life, the universe and everything - I've come up with a...

The possibility of "Perception" being a dimension? Is it possible that perception itself could be a dimension? Now think deeper, not personal perception, but instead the idea of perception in general. Because quiet simply, there are lots of things that exist that we cannot perceive. Just as...

Find the basis and dimension of the following homogeneous system: A = |1 0 2| |x1| |2 1 3| |x2| = [0,0,0] |3 1 2| |x3| My attempt: Solving the coefficient matrix for RREF, I get the identify matrix. So, x1=x2=x3=0 and the only solution is a trivial one. Does that mean...

Hello forum, today in my gr. 12 physics class I had a interesting question which I could not prove at all for the life of me. It is a really simple question, just with a little twist I suppose. As I was driving home from class I was thinking about what I was missing and thought up a solution...

Homework Statement He made some notes, but I'm still confused. The Attempt at a Solution

Hello, I just need to know whether or not surfaces with zero size in the third dimension, 6x8x0, is considered two-dimensional. The surface is there all the time. It has a location in the third dimension, so wouldn't it be a 3D object? I am not sure whether I should call a flat surface (as...

I've recently been learning about relativity and quantum physics. I understand it quite well now but i have a thought experiment that doesn't make sense. If i travel into the future through a high speed train and see the world blown up by some epic disaster, doesn't this mean that this was...

I have the following problem: I'm studying a system of polynomial equations in R^n and I'm looking at the surface which is the solution set of this system. I'm mainly interested in the dimension of this surface at a given point. Now, naively, one would try to compute the Jacobian (of the m...

So, I was just thinking what if the time dimension was a tiny curled up dimension like one of these supposed extra 8 dimensions from string theory ? If it allowed just enough wiggle room for objects to move the idea has some features that appeal to me. - time is symmetrical ; no past , no...

Hi forum, I was having some difficulty in understanding how time as a fourth dimension allows a cube of three spatial dimensions to obtain the properties of a tesseract. Suppose we allow a cube to exist for some interval of time. At the start such a cube would have eight vertices as well as...

I don't really understand what the dimension of a vector space for planes is. Is it 2 or 3 and why? What's the difference between the dimension of the plane as a surface (dimension of all surfaces is 2) and the dimension of plane in a vector space? Also, Wikipedia says that only planes passing...

Homework Statement A fugitive tries to hop a freight train traveling at a constant speed of 6.0 m/s. Just as an empty box car passes him, the fugitive starts from rest and accelerates at a=4.0 m/s2 to his maximum speed of 8.0 m/s. (a) How long does it take him to catch up to the empty box...

Homework Statement Calculate the following integrals: (a) I(n,\alpha) = \int_{0}^{\infty} e^{-\alpha x^2}x^n dx for n whole integers and n \ge 0 Calculate all results till n=5. Tip: First calculate I^2(0,\alpha) and I(1,\alpha) and then use this to calculate n>1. (b) I(n)=\int_{0}^{\infty}...

I suppose it might be appropriate here. I'm not really looking for an answer in pure maths (Topology is waaaaaaaaaaay beyond me), but rather in concept. Now, I can appreciate the properties of a tesseract, how it relates to and derives from the cube (I think of it as either putting 8 cubes...

I have a question in my linear algebra text that asks: Give integers p and q such that Nul A is a subspace of Rp and Col A is a subspace of Rq. What determines these values? Why are the values of p and q different between the Nul space and Col space? The matrix in question is a 3 x 4...

Is it the additional dimension, Time, added to the Euclidean (x,y,z) 3-D space? Or does it even make sense to ask this question? And instead consider 4D space-time (Minkowski Space) as a manifold in which 3D is embedded? I'm confused. :(

Issue with the definition of the Hausdorff dimension Homework Statement http://mathworld.wolfram.com/HausdorffDimension.html" involves a n-dimensional Hausdorff measure of 0. I'm having trouble understanding cases that would give such a value. Homework Equations the Hausdorff...

Homework Statement Find (a) a basis for and (b) the dimension of the solution space of the homogeneous system of equations. x - 2y + z = 0 y - z + w = 0 x - y + w = 0 Homework Equations The Attempt at a Solution (a) [1 -2 1 0] => [1 0 -1 2] [0 1 -1 1] => [0 1...

To remove an electron from a H-like atom, the action of a photon is required. But what is the dimension of a photon, that hits the electron? If it is considered a wave, then is it possible to have just energy without mass, and it is known that the extra energy of the photon is imparted to the...

Dear everybody... Could it be true... that Entanglement is made possible because the forces between the 2 electrons are being folded through a higher dimension, so there would be an explanation for the simultaneous reaction of the 2 electrons even if there`s a astronomical distance between...

Hi, I read that the dimension of an attractor is not necessarily an integer. I then tried to compute the dimension of the attractor defined by the system I’m studying but I found two definitions for the dimension: -The capacity dimension; -The...

Is string theory the replacement of the zeroth dimension? Or could it be thought of as such?

Homework Statement Tarzan is in the path of a pack of stampeding elephants when Jane swings into the rescue on a rope vine, hauling him off to safety. The length of the vine is 27 m, and Jane starts her swing with the rope horizontal. If Jane's mass is 56 kg, and Tarzan's mass is 91 kg, to...

Homework Statement Let S={v1=[1,0,0,0],v2=[4,0,0,0],v3=[0,1,0,0],v4=[2,-1,0,0],v5=[0,0,1,0]} Let W=spanS. Find a basis for W. What is dim(W)? Homework Equations The Attempt at a Solution i know that a basis is composed of linearly independent sets. This particular problem's...

what is 4th dimension? is it just 1 dimensional space? how come time is described as the 4th dimensional path? thanks..