Dimension Definition and 908 Threads

In other words, why is dim[{0}]=0. My math professor explained that since the 0 vector is just a POINT in R2 that the zero subspace doesn't have a basis and therefore has dimension zero. This is not satisfactory. For example, I know R2 has a dimension 2, P_n has dimension n+1, M_(2,2) has...

I have a lab report due tonight. We were working on these in lab and I'm not sure how to tell which one was elastic or inelastic. We had three cases. We had to use the conservation of momentum and kinetic energy formulas.A ) Cart A and B have equal masses. Car B is at rest. Car A strikes car B...

1. The problem statement Given an n-dimensional CW complex on a space X, show that each n-cell is open in X. Definition of CW Complex 1. The cells form a disjoint union of X 2. For each k-cell e, there is a relative homeomorphism : \phi: (D^k, S^{k-1}) \rightarrow (e \cup X^{k-1}...

I've been day dreaming about this idea for a month or two. There are many hypothesises that describe the universe in some higher number of dimensions. As far as I know, this is always an integer number of dimensions. I've been wondering if it is possible for the universe to have a...

New here. I just needed to tell my story somewhere. I'll warn you its not very interesting. Its just a strange thing that happened in my life and I cannot explain. I was detoxing of benzodiazepines at a medical detox center. I had a seizure. As a result I went into a coma or coma like...

So an example was the matrix: A = \left(\begin{array}{cccc} a&a+b\\ b&0\\ \end{array} \right) is a subspace of M2x2. and is the linear combination a*\left(\begin{array}{cccc} 1&1\\ 0&0 \end{array} \right) + b*\left(\begin{array}{cccc} 0&1\\ 1&0 \end{array} \right) Meaning it has...

If S is subspace of R6x6 consisting of all lower triangular matrices, what is the dimension of S? Does anyone know the properties about dimension of lower triangular matrices?

I need to ask a scientific community here if there are any sort of basis for the video and the book mentioned above in the title? once i got to the fifth dimension it all turned to non sense that does not relate in any way to string or M theory. Supposing I am right about what i said...

Is my understanding correct. Gravity is a week force because gravitons escape out of our dimension into other dimensions. If so would gravitons not also be able to re-enter our dimesion. Therefore cancelling out this theory

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

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...

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...

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...

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, ...

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...

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...

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\...

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...

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...

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...

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.

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...

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...

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...

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...

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...

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...

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...

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

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}...

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...

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...

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

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) =...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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