Dual Definition and 354 Threads

I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and $\mathcal L^k(V_1, \ldots, V_k;\ F)$. Define a map $A:V_1^*\times\cdots\times V_k^*\to \mathcal L^k(V_1...

Hello, I'm reading Sean Carroll's Spacetime and Geometry. When discussing dual vectors, he presents the gradient as the "simplest example of a dual vector" in spacetime. This confuses me because I learned the gradient to be an operator which takes a scalar as input and outputs a vector. The way...

Let $$T: X \rightarrow Y$$ be a continuous linear operator between Banach spaces. Prove that $T$ is surjective $$\iff$$ $$T^*$$ is injective and $$im T^*$$ is closed. I've proven a "similar" statement, with $$imT^*$$ replaced with $$imT$$. There I used these facts: $\overline{imT}=...

Dear members I'am stuck in Dual mass spring system. How do we relate acceleration of masses when they are are oscillating relatively with let say amplitudes x1 and x2? Is there any simpler approach available please let me know.thanks.

1st order crossover desired low-pass frequency: 100hz At 8 ohms, that'd be about 12mH. But what happens with a DVC speaker? Each coil get a 12mH inductor? I'd like to think so... Speaker is 8+8 ohms and will be parallelled to a set of 8 ohm extended range speakers showing my amplifier 4 ohms...

Homework Statement Let Hom(V,W) be the set of linear transformations from V to W. Define addition on Hom(V,W) by (f + g)(v) = f(v) + g(v) and scalar multiplication by (af)(v) = af(v. If V is a vector space over a field K, define V* = Hom(V,K). This is called the dual space of V. If...

Hi all. I was hoping I could clarify my understanding on some basic notions of dual spaces. Suppose I have a vector space V along with a basis \lbrace\mathbf{e}_{i}\rbrace, then there is a unique linear map \tilde{e}^{i}: V\rightarrow \mathbb{F} defined by \tilde{e}^{i}(\mathbf{v})=v^{i}...

Who knows a goodbook of physics with a good approach on the star operator? I want to practice with it. Thank you in advance.

I know that photons and other particles of comparatively smaller dimension show dual nature. But why? Can someone help me? :confused:

Hey guys, Science I want to major in either applied mathematics or theoretical physics. I know it sounds cute to most real mathematicians but I love solving integrals (serious) and even in my free time I print out sheets at home and do numerous of weird integral calculus problems. However, I...

Hello all. So i have been trying to simulate a ndg transistor mixer in LTSpice. Here is what I have so far, but I am missing something and I can't figure what it is... Here is the schematic along with the waveforms. For some reason the output will not go over 0.3V (do I have to bias it?)...

Would it be beneficial at all (as regards lean burn and rate of flame propagation) to have two spark plugs, one charged with a high positive voltage, one charged with a high negative voltage, such that the spark jumps between the two plugs? My thinking (which is probably wrong, as it often...

Hi. This is clearly wrong, but I don't know where is the error: ##\langle n\vert = (\vert n \rangle )^* = (a_+^n\vert 0 \rangle )^* = \langle 0\vert a_{-}^n = 0 ##

I've been reading about algebraic geometry lately. I see that a lot of authors use ##V^\vee## to denote the dual space of a vector space ##V##. Is there any particular reason for this? The only reason I could think of is that this notation leaves us free to use ##R^*## to denote the units of...

I am making a small vertical savonius type wind turbine and would like some general rule guidelines. I am using 14awg magnetic wire--70 turns (2"oval) and ceramic block 1" x 2" x.375" magnets. I am hoping to get the most power possible at lower wind speeds. Is there a rule that states a...

Can the same force increase both the translational and rotational kinetic energy of a rigid body? If yes then the work done by the force equals the sum of the increase in the translational and rotational kinetic energy?

I am a high school senior going into university this fall. I have completed many general education credits and I am going in as a freshman with 68 credits already completed. Because of this, I am interested in instead of taking 2 1/2 or 3 years in university I would be willing to take 4 years...

The problem statement, all variable Let ##\phi_1,...,\phi_n \in V^*## all different from the zero functional. Prove that ##\{\phi_1,...,\phi_n\}## is basis of ##V^*## if and only if ##\bigcap_{i=1}^n Nu(\phi_i)={0}##. The attempt at a solution. For ##→##: Let ##\{v_1,...,v_n\}## be...

Homework Statement . Let ##A \in \mathbb C^{m\times n}##. Prove that tr##(A^*A)=0## if and only if ##A^*A=0## (here ##0## obviously means the zero matrix). The attempt at a solution. By definition of the trace of a matrix, the implication ← is obvious. I am having problems proving...

Can someone please provide me with some research topic ideas for dexa?

Why is it the case that dual field tensors, e.g. \widetilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho \sigma}, aren't being included in the Lagrangian? For example, one doesn't encounter terms like -\frac{1}{4}\widetilde{F}^{\mu\nu}\widetilde{F}_{\mu\nu} in QED or...

Hello everyone, I have recently read a puzzling statement on my Electromagnetism (Chapter on Special Relativity) material regarding the Field Strength Tensor, F^{\mu\nu}, and its dual, \tilde{F}^{\mu\nu}. Since I've been thinking about this for a while now, and still can't understand it, I...

Hello. I've just read about natural identifications of exterior powers with spaces of alternating maps, etc here: Some Natural Identifications However, I have problems showing that the following operations give the same space: $$V \rightarrow \Lambda_p V \rightarrow (\Lambda_p V)^* \cong...

Homework Statement Find Vo as a function of Vs Homework Equations The Attempt at a Solution My thought process so far is to do KCL and find Vo as a function of Vn2, then Vn2 as a function of Vo1, then Vo1 and Vo as a function of Vn1, then Vn1 as a function of Vs, finally giving me...

Hi, if ket is 2+3i , than its bra is 2-3i , my question is 2+3i is in Hilbert space than 2-3i can be represented in same hilbert space, but in books it is written we need dual Hilbert space for bra?

Hi, I am looking for a compact (possibly a on-chip solution) high efficiency DC-DC dual converter powered by a USB 2 port. The aim is to supply a dual voltage to a logic circuit for driving audio signals from/through different channels, as in an audio switch I.e. MAX4910 or 4912. Any idea...

Some sources I have checked define the Hodge dual of a form \omega \in \Omega^p as the object such that \forall \eta \in \Omega^p: \eta \wedge \omega^\star = g(\eta,\omega) \textrm{ Vol} (where "Vol" is a chosen volume form). I can see that there can be only one form with such a solution...

I'm in a Second Course in Linear Algebra this semester, and we've just been introduced to the idea of a dual space, dual vectors and briefly to a double dual space. I completely understand how all of these things work and how they're defined, but I don't understand why we care. I've been...

I desire both these skill sets, but I was wondering if computer engineering would be more appropriate, as I am under the impression that it is a mix of both CS and EE, but with a focus on computer hardware. If someone could enlighten me to the main differences between these two paths that would...

I searched the forums, and I can't find anywhere where someone asked this question point blank. I may be completely off base so I apologize in advance if that is the case. In quantum mechanics, an inner product is formed as the bra-ket <φ|ψ>. We are told that vectors are represented by the...

Ok, so I have many questions about light. My basic understanding can be seen here: http://answers.yahoo.com/question/index?qid=20140117062517AAeH0Dh My first question is, consider transverse waves in water. As the wave moves out, the water molecules move up and down forming the crests and...

Hi, All: Let X be a Reeb vector field, and let ω be a 1-form dual to X. Is ω necessarily a contact form? I know if we have a contact 1-form θ , and Zθ is the Reeb field associated with θ , then from the definition of Reeb field, we have θ (Zθ)=1, which mostly means that θ is...

I am trying to calculate the dual form of an SVM optimisation problem: Dual Form Optimsation Problem In my algorithm, I have a vector of alphas, vector of target outputs, and a Kernel matrix computing upfront. However, I am stuck as to what indices alpha and j should be taking here...

Hi, I was just curious, I am no EE, I am an ME and I am trying to make something smaller if you have read my other posts you might know what's it all about, anyways I've been reading a lot about single supply op amps and that's probably the best choice to reduce the amount of power my circuit...

After taking a couple classes on programming (I know, CS isn't just programming) and an upper division intro to combinatorics, I'd like to get more into the area of discrete mathematics. I hope to pursue a PhD at some point, but I'm not sure how strong of a school I can get into since currently...

Homework Statement There are three qutrits which are connected by quantum entanglement: |\Psi\rangle=\frac{1}{\sqrt{3}}\left( {|021\rangle+|102 \rangle+ |102 \rangle} \right) How can I to describe \langle\Psi| ? \langle\Psi|=\frac{1}{\sqrt{3}}\left( {\langle ... |+\langle ... | +...

Greetings PF. I was thinking about going for a dual masters degree with one in physics and one in mathematics. Note that I'm not American, I'm currently in Europe and as such I'd guess it's almost like a double major. To me it felt like such a couple of degrees were a natural choice, but...

I have been reading an introductory book to General Relativity by H Hobson. I have been following it step by step and now I am stuck. It is stated in the book that: "It is straightforward to show that the coordinate and dual basis vectors themselves are related... "ea = gabeb ..." I have...

Hi everyone, :) Here's a question and I'll also write down the answer for which I got zero marks. :p I would really appreciate if you can find where I went wrong. Question: Let \(\phi,\,\psi\in V^{*}\) be two linear functions on a vector space \(V\) such that \(\phi(x)\,\psi(x)=0\) for all...

How would one go about attaining a dual doctorate in math and physics? Would I be able to do both at the same time? Do people usually go for the doctorate after bachelors? What colleges allow such a program? thanks.

hi what would happen if i modulated two wavelengths with the same intensity modulation frequency? for example if i had 2.0um light and 1.5um light and modulated them both with the same frequency, would sidebands develop around each wavelength? or would they interfer in a more complicated...

Hello, Whenever reading about op amps, I have come across two terms, 'dual supply' and 'single supply'. Whenever referring to dual supply, I have seen +/- next to the indicated value of the supply, such as +/- 5 volts. Though, whenever single supplies are discussed, there is no negative...

Hi, I'm trying to get my head around the Hodge dual and how it exactly works. In the book "Gauge Fields, Knots and Gravity" by John Baez and Javier P. Muniain they define: \begin{equation} \omega \wedge * \mu = \langle \omega , \mu \rangle \mathrm{vol} \end{equation} for two p-forms...

Hi, I have a related question to one I just posted: Is H^DaggerH a dual vector space, or perhaps a dual vector field? Could H^dagger exist independently of H, or are they considered a kind of pair?

Just a question. A particle and a corresponding antiparticle (e.g. electron and positron) can annihilate by mutual interaction, producing energy (photons). If the process of mutual annihilation occurs necessarily between particles and antiparticles (matter and antimatter), we would expect...

Right now I'm going into my junior year of high school, and college has been on my mind a lot. I just finished AP Chemistry and I'm going to be taking AP Physics B next year and I am interested in both sciences very much, I was wondering if some colleges offer a double major program for...

Stats after the fall semester (the time I will be applying for graduate school for Physics) Best case scenario I will have a 3.4 cumulative GPA, 3.85 Physics GPA, and a 3.0 Math GPA Dual Mathematics and Physics major, took 5 years to complete. 1st - 3rd year grade were par or less...

Hello everyone, looking around I have faith that the members of this forum will be able to point me in the right direction, and I apologize if it's more basic than I'm giving it credit for. I'm an experimental researcher in rock mechanics, but I've always been fascinated by elasto-dymamic...

As a soon to be physics major I am about to purchase a laptop. However, I can't decide what I want. Should I go for a laptop with a dual core processor with a higher clock rate or a quad core processor with a slower clock rate. The laptop with the quad core is a bit out of my price range but I...

I'm currently studying a BEng (Mechanical) and was wondering about dual major options at my university. The two other majors that I'm looking at are Mechanical and Aerospace, and Mechatronic. I'm also considering doing a Masters of in either Electrical Engineering, Electricity Market or Systems...