Inner product Definition and 312 Threads

Homework Statement I have a linear transformation ##\mathbb{R}^3 \rightarrow \mathbb{R}^3##. The part that asks for a basis of eigenvectors I've already solved it. The possible eigenvectors are ##(1,-3,0), (1,0,3), (\frac{1}{2}, \frac{1}{2},1) ##. Now the exercise wants me to show that there is...

Homework Statement Prove the following form for an inner product in a complex space V: ##\langle u,v \rangle## ##=## ##\frac 1 4####\left| u+v\right|^2## ##-## ##\frac 1 4####\left| u-v\right|^2## ##+## ##\frac 1 4####\left| u+iv\right|^2## ##-## ##\frac 1 4####\left| u-iv\right|^2## Homework...

Homework Statement Can someone please check my working, as I am new to Einstein notation: Calculate $$\partial^\mu x^2.$$ Homework Equations 3. The Attempt at a Solution [/B] \begin{align*} \partial^\mu x^2 &= \partial^\mu(x_\nu x^\nu) \\ &= x^a\partial^\mu x_a + x_b\partial^\mu x^b \ \...

Homework Statement T/F: If ##T: \mathbb{R}^n \rightarrow \mathbb{R}^m## is a linear transformation and ##n>m##, then the function ##\langle v , w \rangle = T(v) \cdot T(w)## is an inner product on ##\mathbb{R}^n## Homework EquationsThe Attempt at a Solution The first three axioms of the inner...

I got the following derivation for some physical stuff (the derivation itself is just math) http://thesis.library.caltech.edu/5215/12/12appendixD.pdf I understand everything until D.8. So in the equation ε is a symmetric matrix and δx(t) is just the difference between two points. After D.7...

Homework Statement 2. and 3. Relevant equations and the attempt at a solution We find the L^2 projection as such: <b_j , e_j > , where e_j is orthonormal basis j. Now set b_j = < x^2 , e_j > for 1 \leq j \leq 3 .

Homework Statement Consider the vector space of all continuous functions on the interval C[-1,1]. That is V = C[-1,1] show that <f(x),g(x)> = ∫(-1,1) x2f(x)g(x)dx defines an inner product on C[-1,1]. I have shown <f,g> = <g,f>, <kf,g> = k<f,g>, <f+g,h> = <f,h> + <g,h> and I am trying to...

I am trying to learn Geometric Algebra from the textbook by Doran and Lasenby. They claim in chapter 4 that the geometric product ab between two vectors a and b is defined according to the axioms i) associativity: (ab)c = a(bc) = abc ii) distributive over addition: a(b+c) = ab+ac iii) The...

Homework Statement Consider the vector space R2 with the standard inner product given by ⟨(a, b), (c, d)⟩ = ac + bd. (This is just the dot product.) PLEASE SEE THE ATTACHED PHOTO FOR DETAIlS Homework Equations T(v)=AT*v The Attempt at a Solution I was able to prove part a. I let v=(v1,v2)...

Hello I found a bug in my code and can't figuring out the error. I tried debugging by showing the output of each variable step by step but I can't find my error. Here is what I have and what I want to do: I have a matrix A: 0000 0101 1010 1111 And I have a matrix B: 10000 21000 30100 41100...

(Not an assigned problem...) 1. Homework Statement pg 244 of "Mathematical Methods for Physics and Engineering" by Riley and Hobson says that given the following two properties of the inner product It follows that: 2. Attempt at a solution. I think that both of these solutions are...

Let ##V## be a quaternionic vector space with quaternionic structure ##\{I,J,K\}##. One can define a Riemannian metric ##G## and hyperkahler structure ##\{\Omega^{I},\Omega^{J}, \Omega^{K}\}##. Do this inner product $$\langle p,q \rangle :=...

Can anyone show me explicit examples of Hyperhermitian inner product?

I'm trying to prove the following relation $$\langle\psi\lvert \hat{A}^{\dagger}\rvert\phi\rangle =\langle\phi\lvert \hat{A}\rvert\psi\rangle^{\ast}$$ where ##\lvert\phi\rangle## and ##\lvert\phi\rangle## are state vectors and ##\hat{A}^{\dagger}## is the adjoint of some operator ##\hat{A}##...

Hi, I have this question: in the context of linear algebra, would it be correct to say that a metric is a kind of inner product?

Hi all, The basis vectors are defined as 1x3 matrices, how can the result be a 3x3 matrix? How can the result of a dot product be a 3x3 matrix, I'm stumbled, how can I evaluate this? A inner product returns a scalar, and now it returns a 3x3 matrix, please help. Thanks.

I would like to gain a more formal mathematical understanding of a construct relating to spinors. When I write down Dirac spinors in the Weyl basis, I see why if I multiply the adjoint (conjugate transpose) of a spinor with the original spinor I don't get a SL(2,C) scalar. It just doesn't work...

This is my question: What is the largest m such that there exist v1, ... ,vm ∈ ℝn such that for all i and j, if 1 ≤ i < j ≤ m, then ≤ vi⋅vj = 0 I found a couple of solutions online. http://mathoverflow.net/questions/31436/largest-number-of-vectors-with-pairwise-negative-dot-product...

Homework Statement Suppose R and Q are two quantum systems with the same Hilbert space. Let |i_R \rangle and |i_Q\rangle be orthonormal basis sets for R and Q . Let A be an operator on R and B an operator on Q . Define |m\rangle := \sum_i |i_R\rangle |i_Q\rangle ...

Homework Statement Let A be an nxn matrix, and let |v>, |w> ∈ℂ. Prove that (A|v>)*|w> = |v>*(A†|w>) † = hermitian conjugate Homework EquationsThe Attempt at a Solution Struggling to start this one. I'm sure this one is likely relatively quick and painless, but I need to identify the trick...

Homework Statement Consider a qubit in the state |v> ∈ ℂ^2. Suppose that a measurement of δn is made on the qubit. Show that the probability of obtaining the result "+1" in the measurement is equal to 0 if and only if |v> and |n,+> are orthogonal. Homework Equations Inner product axioms |v>|w>...

Homework Statement Homework EquationsThe Attempt at a Solution I was able to do the second part of part a using integration by parts. But I am having no luck for the first part, proving that the inner product is positive definite. Pointers are appreciated!

Homework Statement Show that ∫ ψ1(x)*ψ2(x) dx = ∫φ1(k)*φ2(k) dk (Where the integrations are going from -∞ to ∞) Homework Equations 1. Plancherel Theorem: ψ(x) = 1/√2π∫φ(k)eikx dk ⇔ φ(k) = 1/√2π∫ψ(x)e-ikx dx The Attempt at a Solution It is clear that Plancherel's theorem must be used to...

Homework Statement I'm having trouble understanding the definition of a complex inner product. Let λ ∈ ℂ So if we have <λv|w> what does it equal to? Does it equal λ*<v|w> where * is the complex conjugate?Are all these correct: <λv|w> = λ*<v|w> <v|λw> = λ<v|w> <v|w> = (<w|v>)* <v|w> = Σvw...

Homework Statement Prove that <v|0>=0 for all |v> ∈ V. Homework EquationsThe Attempt at a Solution This is a general inner product space. I break it up into 2 cases. Case 1: If |v> = 0, the proof is trivial due to inner space axiom stating <0|0> = 0. Case 2: If |v> =/= 0 then: I use <v|0>...

Hi Guys, that's what i got <x,z>=<y,z> <x,z>-<y,z>=0 <x,z>+<-y,z>=0 <x-y,z>=0 x-y = [0,2,0] <2*[0,1,0],Z>=0 2<[0,1,0],z> = 0 <[0,1,0],z>=0 So 'im stuck at that. Any ideas?

Homework Statement Given basis |x>,|y>,|z> such that <x|x> = 2,<y|y> = 2,<z|z> = 3,<x|y> = i, <x|z> = i, and <y|z> = 2. Build an orthonormal basis|x'>,|y'>,|z'>. Each of the new basis vectors should be expressed in terms of the old ones multiplied by coefficients. Homework Equations |x'> =...

Hi, this is a rather mathematical question. The inner product between generators of a Lie algebra is commonly defined as \mathrm{Tr}[T^a T^b]=k \delta^{ab} . However, I don't understand why this trace is orthogonal, i.e. why the trace of a multiplication of two different generators is always zero.

Hello everyone, I would like to know if anyone knows what is the inner product for vector fields ##A_\mu## in curved space-time. Is it just: $$ (A_\mu,A_\mu)=\int d^4x A_\mu A^\mu =\int d^4x g^{\mu\nu}A_\mu A_\nu $$ ? Do I need extra factors of the metric? Thanks!

Homework Statement Using the equations that are defined in the 'relevant equations' box, show that $$\langle n' | X | n \rangle = \left ( \frac{\hbar}{2m \omega} \right )^{1/2} [ \delta_{n', n+1} (n+1)^{1/2} + \delta_{n',n-1}n^{1/2}]$$ Homework Equations $$\psi_n(x) = \left ( \frac{m...

Let X be an Inner Product Space. If for every closed subspace M, M^{\perp \perp} = M, then X is a Hilbert Space (It's complete). Hint: Use the following map: T : X \longrightarrow \overset{\sim}{X}: T(y)=(x,y)=f(x) where (x,y) is the inner product of X. Relevant equations: S^{\perp} is always...

(I hope this post goes in this part of the forum) Hi, I was wondering if someone could help me with the following: I have a (1+1) scalar field decomposed into two different sets of modes. One set corresponds to a Minkowski frame in (t,x) coordinates, the other to a Rinder frame in conformal...

Let u = [1, 2, 3, -1, 2]T, v = [2, 4, 7, 2, -1]T in ℝ5. Find a basis of a space W such that w ⊥ u and w ⊥ v for all w ∈ W. I think the question is quite easy. Given this vector w in the space W is orthogonal to both u and v. I can only think of w being a zero vector. But would this be too...

Hi all, I am a final year maths student and am doing my dissertation in the finite element method. I have gotten a little stuck with some parts though. I have the weak form as a(u,v)=l(v) where: $$a(u,v)=\int_{\Omega}(\bigtriangledown u \cdot\bigtriangledown v)$$ and $$l(v)=\int_\Omega...

Homework Statement This question has two parts, and I did the first part already I think. If B = {u1, u2, ..., un} is a basis for V, and ##v = \sum_{i=1}^n a_i u_i## and ##w = \sum_{i=1}^n b_i u_i## Show ##<v,w> = \sum_{i=1}^n a_i b_i^* = b^{*T}a## Here's how I did it: ##<v,w> =...

Homework Statement Let ##V## be an inner product space and let ##V_0## be a finite dimensional subspace of ##V##. Show that if ##v ∈ V## has ##v_0 = proj_{V_0}(v)##: ||v - vo||^2 = ||v||^2 - ||vo||^2 Homework Equations General inner product space properties, I believe. The Attempt at a...

I'm interested in what people know about the application of inner product structures (usually reserved for QM) to diff equations describing classical physics, in particular non- hermitician diff operator of the Fokker-Plank equation. Thanks.

Homework Statement Let V be a finite-dimensional real vector space with inner product <⋅,⋅> and L: V → R a linear transformation. Show that there exists a unique vector a ∈ V such that L(x) = <a,x>. Homework Equations Hey everyone, so I'm a physics student who had to choose a few electives in...

Homework Statement [/B] Let Z be any 3×3 orthogonal matrix and let A = Z-1DZ where D is a diagonal matrix with positive integers along its diagonal. Show that the product <x, y> A = x · Ay is an inner product for R3. Homework Equations None The Attempt at a Solution I've shown that x · Dy is...

Homework Statement http://postimg.org/image/lgphyvggz/ Homework Equations The Attempt at a Solution can someone explain where that transpose came from in (3.3)?

Hi all, I'm trying (and failing miserably) to understand tensors, and I have a quick question: is the inner product of a rank n tensor with another rank n tensor always a scalar? And also is the inner product of a rank n tensor with a rank n-1 tensor always a rank n-1 tensor that has been...

Homework Statement Suppose \vec{u}, \vec{v} and \vec{w} are vectors in an inner product space such that: inner product: \vec{u},\vec{v}= 2 inner product: \vec{v},\vec{w}= -6 inner product: \vec{u},\vec{w}= -3 norm(\vec{u}) = 1 norm(\vec{v}) = 2 norm(\vec{w}) = 7 Compute...

While studying Yang-Mills theory, I've come across the statement that there exists a positive-definite inner product on the lie algebra ##\mathfrak g## iff the group ##G## is compact and simple. Why is this true, and how it is proved?

Hey guys I'm new here and I've been using MathType for all on-screen math. For some reason PF doesn't recognize the built-in Dirac's bra-ket notation. (i.e. <\psi|) So I've included my equations and solution in the format of images, hopefully it isn't a problem. Homework Statement Proof...

Homework Statement Hi, I am having difficulty with the following proof: Let V be an inner product space (real of dimension n) with two inner products in V, <,> and [,]. Prove that there exists a linear mapping on V such that [L(x),L(y)] = <x,y> for all x,y in V. I am stuck as to where...

I was reading in my textbook that the scalar product of two complex vectors is also complex (I assuming this is true in general, but not in every case). However for the general definition (the inner product), each element of one of the vectors needs to be its complex conjugate. I learned this...

Hey! :o We know that: $$(x,x)=0 \Rightarrow x=0$$ When we have $\displaystyle{(x,y)=0}$, do we conclude that $\displaystyle{x=0 \text{ AND } y=0}$. Or is this wrong? (Wondering)

So the complex exponential Fourier series form an orthonormal basis for the space of functions. A periodic function can be represented with countably many elements from the basis, and an aperiodic function requires uncountably many elements. Given a signal, we can find the coefficients of the...

Homework Statement You are given that with x = (x1,x2), y = (y1,y2), the formula (x,y) = [x1 x2] [2 1;1 2] [y1;y2] (where ; represents a new row). is a inner product for the vectors in R2 Using this inner product, find a unit vector perpendicular to the vector (1,1) Homework Equations The...

Hi guys, I'm not sure how to evaluate this inner product at step (3.8) I know that: ##\hat {H} |\phi> = E |\phi>## <E_n|\frac{\hat H}{\hbar \omega} + \frac{1}{2}|E_n> <E_n| \frac{\hat H}{\hbar \omega}|E_n> + <E_n|\frac{1}{2}|E_n> I also know that ##<\psi|\hat Q | \psi>## gives the...