Dimension Definition and 881 Threads

Prove: Let $A$ be an $n$ by $n$ matrix of rank $r$. If $U={}\left\{X \in M_{nn}|AX=0\right\}$, show that $\dim U=n(n-r)$. Proof: Clearly, $\dim \operatorname{null}A=n-r$, and let $X=[c_1,c_2,...,c_n]$ where $c_i \in \Bbb{R}^n$ are column vectors. Then since $AX=0$, using block multiplication...

Homework Statement Find a basis for and the dimension of the solution space of the homogenous system of equations. 2x1+2x2-x3+x5=0 -x1-x2+2x3-3x4+x5=0 x1+x2-2x3-x5=0 x3+x4+x5=0 Homework EquationsThe Attempt at a Solution I reduced the vector reduced row echelon form. However the second row...

Homework Statement I am curious as to how the second line in the equation is equal to the third line in the equation. The book my class is using is Taylor and it just skips so many steps. What happens to the sign, I know this must relate to euler in some way I am just not sure how. Thank you...

Hello. I need to determine the diameter of an exhaust pipe coming out of a 14" air duct. The exhaust would be perpendicular to the duct and connect to a vertical stack which exhausts to atmosphere. Supply air is 5000scfm at 2.5psi and 120F going into a gas heater which adds 400scfm air to the...

Just a simple question -- can the dimension of coupling constant be a rational number or should it always be an integer? The question arose when I was trying to construct a Lagrangian with an interaction term involving two spin-1 particles and a fermion. The dimensions add up to 7/2, which...

I know about the tessaract and I'd like to understand more about it from a Euclidean perspective so I may translate it algebraically.

Do you just cube a variable? How does it work? Has this been figured out yet? Any thoughts if it hasn't? Any applications if it has?

please point out the flaws/add to my reasoning here: it is known that energy is quantized, but could that be for just the 3 dimensions we're able to experience? Is it illogically to propose that energy might not be quantized when looking at all of the dimensions together? this could explain...

Homework Statement If x and t represent position and time, respectively, and A and B are constants in x = At2{1-exp(-t2/B)}, what are the dimensions of A and B? 2. The attempt at a solution I just slept through my first class and found that I have many homework problems related to...

In the process of descending from a higher dimension to a lower one, there must be more than one factor that could not be solved (or descend.) My little own experiment.. http://mr-none.meximas.com/public_html/pic/1.JPG Steps: 1, wrap a plastic bag around a basketball. 2, draw a "triangle" with...

Homework Statement I'm trying to find the point at which FG attracting to the moon = FG attracting to the earth... 5.98e24 kg = Earth Mass 7.35e22 kg = Moon Mass 3.84e8 m = Distance from Earth to Moon Deductions... a = 3.84e8 - b b = 3.84e8 - a a+b = 3.84e8 Homework Equations g = G m / r2...

I know this has been mulled over time and time again in different threads, so I will keep it short. Which statement is more consistant with reality: 1. A dimension = all existing and non-existing points along an axis. 2. A dimension = all existing, but only existing points along an axis...

I am still learning of Kaluza-Klein theory. Is KK theory applying to quantum mechanics? How quantum non-locality can be represented by KK theory?

Often ignored, but turned out to be a problem when trying to compute the commutator of position and spin. Pauli matrices clearly acting on two dimensional vectors while position on infinite dimensional vectors. But a system is described as a single state vector in Dirac notation. A system can of...

Ok, so I was just in physics class today and we were talking about special relativity... anyways, the instructor referred to how the universe is expanding, and so much so that there are places we could never ever get to because it's expanding fasterling than the speed of light. Anyways, this...

hi guys, i am new here, not even sure this is right place to put this questions, I always understood time as a 4 dimension, just a diferent reference when compared with x,y and z. but is it possible that we r trying to see it in a different way? is it possible that time itself have its own x y...

Homework Statement can someone explain why we are interested in forming dimensionless products and why only n-j of them should be formed from the problem's variables? Homework Equations Step 1: List the variables in the problem Step 2: Express each of the variables in terms of basic...

How can time be a dimension, if it is a human construct, to measure the rate of change? I'm only in high school, sorry if this is a silly question

If you have a vector space S be spanned by vectors (x1,y1,z1), (x2, y2, z2), (x3, y3, z3) and T spanned by (x1,y1,z1), (x2, y2, z2), (x3, y3, z3). How would you find the basis and dimension of the intersection of S and T . (x,y,z can be any value) Do I go about it like this? a(x1,y1,z1)+b(x2...

Homework Statement Given a conservative force, how can we obtain the change in potential energy? Given a potential energy function, how can we determine the associated conservative force? One dimensional.Homework Equations Fx = -du/dx ΔU = -∫ F dx The Attempt at a Solution I know I can...

Find dimension and basis of the set of all points in R5 whose coordinates satisfy the relation x1 +x2 +x3 +x4 =0. shouldn't there be 5 vectors to satisfy the basis since they are asking about R5? but the relation only has x1, x2, x3, and x4. or would my matrix just look like this 1 0 0 0 0 0...

Can someone tell me what is the "dimension" of a model? For example, the dimension of a saturated model. thanks

Homework Statement If the electric motor of problem 3.66 is to be mounted at the free end of a steel cantilever beam of length ##5## m and the amplitude of vibration is to be limited to ##0.5## cm, find the necessary cross-sectional dimension of the beam. Include the weight of the beam in the...

Homework Statement is there a 4th dimension. if yes can anyone explain it ?Homework EquationsThe Attempt at a Solution 4th dimension includes time also am i right ? which is X,Y,Z + time

Dirac matrices satisfy the relations: \gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2g^{\mu\nu} I would like to understand why the dimension of this algebra in 3+1 dimensions is 4. If we're looking for all possible sets {\gamma^0,\gamma^1,\gamma^2,\gamma^3} of 4x4 matrices that satisfy this, how...

Let $$\mathbb{Q}(\sqrt{2},\sqrt{3})$$ be the field generated by elements of the form $$a+b\sqrt{2}+c\sqrt{3}$$, where $$a,b,c\in\mathbb{Q}$$. Prove that $$\mathbb{Q}(\sqrt{2},\sqrt{3})$$ is a vector space of dimension 4 over $$\mathbb{Q}$$. Find a basis for $$\mathbb{Q}(\sqrt{2},\sqrt{3})$$. I...

Assuming that the double slit experiment creates interference patterns when electrons interact with themselves (or 'other' selves) from a parallel 'space' - one wonders if one can 'image', or at least gain more information about that other 'space' (dare I use the word 'dimension'?) in which the...

Homework Statement Hi, I have to prove that there's no exist a generating set for "x" with less of "n" vectors when "n" is the dimension of the basis of "x" Homework Equations is there a span(x) whith dimension m? when m<n and n is the dimension of the basis The Attempt at a Solution...

Is it possible practically to measure Hausdorff dimension of the surface of the Brain or the broccoli? For a broccoli Hausdorff dimension is equivalent to the so called box counting dimension, which is far more practical? I think, following the original definition Hausdorff dimension it is quite...

Homework Statement A bar of length ##L## has an initial temperature of ##0^{\circ}C## and while one end (##x=0##) is kept at ##0^{\circ}C## the other end (##x=L##) is heated with a constant rate per unit area ##H##. Find the distribution of temperature on the bar after a time ##t##. Homework...

I have to do a project on the dimension theory buy i can't find any info on it. This is the wikipedia page and if you open it you can see its nonsense: http://en.wikipedia.org/wiki/Dimension_theory so please could i get some info on the dimension theory (the mathenatical theory) AM i right...

Hello, I want to have a basic understanding. When we speak 10 dimension in String Theory, do we mention: (a) The six dimension of Calabu Yau manifold and (4) the four dimension space-time?

We can only engage in discussion of physics through the interface of semantics and -more broadly- linguistics as a whole. No-one has yet devised a method to relate in complete and accurate detail physical phenomenon via mathematical notation alone. Terms have to be given a human understanding...

I have a couple questions I have been thinking about: Say that there is a two dimensional, three dimensional, and four dimensional spatial worlds lined up. If I poked my finger through the two dimensional world they would see a circle. If a fourth dimensional person poked their finger through...

Hi, I´m not sure if my way of tackling a question, probably it's a trivial problem, but it's important for me to get it right so any help will be greeted. The question is as follows: Problem: consider a particle in a one-dimensional system. The wave function ψ(x) is as follows: ψ(x)= 0...

Hofstadter's Butterfly (http://en.wikipedia.org/wiki/Hofstadter%27s_butterfly) is described as a fractal, and in http://physics.technion.ac.il/~odim/hofstadter.html it is stated that when the quantum flux is an irrational number of units, then it is a Cantor set, which makes it (by...

Homework Statement I have been given a complex function I have been given a complex function \widetilde{U}(x,t)=X(x)e(i\omega t) Where X(x) may be complex I have also been told that it obeys the heat equation...

Homework Statement Consider the set of vectors S= {a1,a2,a3,a4} where a1= (6,4,1,-1,2) a2 = (1,0,2,3,-4) a3= (1,4,-9,-16,22) a4= (7,1,0,-1,3) Find the dimension of the subspace Span(S)? Find a set of vectors in S that forms basis of Span(S)? Homework Equations dimension of V = n in Rn...

Homework Statement Determine whether set S = {2a,-4a+5b,4b| aε R ^ bε R} is a subspace of R3? If it is a subspace of R3, find the dimension? Homework Equations dimension= n if it forms the basis of Rn, meaning that its linear independent and span(S) = V The Attempt at a...

Hello to everybody. I had brought up the topic of extra spatial dimension in the math sector of the forum and was directed to the physics section for a question to a second temporal dimension. My questions are the following. would a second temporal dimension run counter to the current arrow...

Hey guys here is my question. 1D = Line 2D is a plane. 3D is space. So shouldn't a fourth dimension be something else? Is there really such a thing as fourth dimensional space. There is no such thing as a 3 dimensional plane. Though you can have a 2D plane in 3D space. Next question leads to...

Homework Statement A projectile is launched at a 60° angle above the horizontal on level ground. The change in its velocity between launch and just before landing is found to be Δv→ = v→landing _ v→launch = -20 y^ m/s . What is the initial velocity of the Projectile ? What is its final...

Hi everyone. I have a very quick question. Can someone tell me how to compute the energy dimensions of an n-point Green function. Consider for example a \lambda\phi^4 scalar theory. I know that the dimensions of an n-pt Green function are 4-n (or something like that). How do I prove it? Thanks

in order to explain the big bang theory and the expansion of space itself, people often draw upon the analogy of blowing air into a balloon and the 2 dimensional surface of the balloon expanding. isn't the 2-D balloon surface expanding in the third dimension, since the volume of the balloon is...

if V = C_x for some x belongs to V then show deg(u_L) = dim(V) here, L: V -> V linear operator on finite dimensional vector space C_x = span {x, L(x), L^2(x),....} u_L = minimal polynomial my thought: since C_x = V, it spans V. if it spans v, then degree of minimal poly that we somehow...

Homework Statement if V = C_x for some x belongs to V then show deg(u_L) = dim(V) here, L: V -> V linear operator on finite dimensional vector space C_x = span {x, L(x), L^2(x),....} u_L = minimal polynomial Homework Equations The Attempt at a Solution since...

I've been thinking about this problem of how we could possibly visualize the fifth dimension. The fourth dimension is easy enough as all you have to do is view a 3D projection of the object as it moves through 3D space. If you look at animations of the projection of a 4D hypercube you'll know...

Suppose that the position vector for a particle is given as a function of time by vector r (t) = x(t)i + y(t)j, with x(t) = at + b and y(t) = ct^2 + d, where a = 1.70 m/s, b = 1.05 m, c = 0.126 m/s2, and d = 1.12 m. So, I evaluated x(t) and y(t) at each time: x(t): 1.70*2.05+1.05 which...

Hello all, I have these two sets (I couldn't use the notation {} in latex, don't know how). V is the set of matrices spanned by these 3 matrices written below. W is a set of 2x3 matrices applying the rule a+e=c+f \[V=span(\begin{pmatrix} 1 &1 &1 \\ 1 &3 &7 \end{pmatrix},\begin{pmatrix} 0 &0...

Hey there peers :smile: I'm really keen on explaining science to others and I've been practicing it with my classmates lately, so I thought why not give it a try and make my first ever youtube video about a seemingly hard concept which is a space of 4 dimensions. I made this post just because...