Inner product Definition and 305 Threads

I'm on a course which is currently introducing me to the concept of Hilbert spaces and the professor in charge was giving examples of such spaces. He ended by considering V, the space of polynomials with complex coefficients from \mathbb{R} to \mathbb{C}. He then, for f,g\in V, defined...

Question from my last LA.II assignment. I have no idea what to with it. It looked simple but now I think I don't even understand the question. Homework Statement Consider the Inner Product Space P2 with the evaluation inner product at points x0=-1, x1=0, and x2=2, and consider the subspace...

Hi, I read that the general form of an inner product on \mathbf{C}^n is: \langle \vec{x} , \vec{y} \rangle = \vec{y}^* \mathbf{M} \vec{x} I see that it has what it takes to be an inner product, but it seems quite hard to demonstrate that this is the general form. Is there such a...

Some one please help me how to prove the following: \dot{A}:B + A:\dot{B}=A^{\nabla J}:B+A:B^{\nabla J} A and B are II order tensors and : represents the inner product.

Suppose you have an inner product space V (not necessarily finite dimensional; so it could be an infinite dimensional Hilbert space or something). Fix a vector \Phi in this space. Given an arbitrary vector \Psi \in V, can I write it as \Psi = \Psi^{\parallel} + \Psi^{\perp}, where...

Hey all, This might seem like a stupid question, and this might not be the correct forum, but hopefully someone can clarify it really easily. I often have seen two definitions of an inner product on a vector space. Firstly, it can be defined as a bilinear map on a \mathbb F-vector space V...

Homework Statement Let V be a finite-dimensional real inner product space with inner product < , >. Let L:V->R be a linear map. Show that there exists a vector u in V such that L(x) = <x,u> for all x in V. 2. The attempt at a solution It seems really simple but I just can't phrase...

An inner product space is often simply described as a vector space with the addition of an inner product, but when it comes to the formal definition, the basefield seems to always be restricted to the fields of real and complex numbers. The Wikipedia article on inner product spaces remarks that...

Homework Statement Homework Equations know dot product The Attempt at a Solution [SIZE="5"]PART A [SIZE="5"]PART B not sure what's it asking for help would be great

Homework Statement Show that \int {{f^*}(x)g(x) \cdot dx} is an inner product on the set of square-integrable complex functions. Homework Equations Schwarz inequality: \left| {\int {{f^*}(x)g(x) \cdot dx} } \right| \le \sqrt {\int {{{\left| {f(x)} \right|}^2} \cdot dx} \int {{{\left|...

I've been trying to learn more about tensors with the help of this website, http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf, but its explanation on one little part about vectors has me puzzled. It states that an inner product of a vector S and a dyad expressed...

Homework Statement 3. If z is any fixed element of an inner product space X, show that f(x) = <x,z> defines a bounded linear functional f on X, of norm ||z||. 4. Consider Prob. 3. If the mapping X --> X' (the space of continuous linear functionals) given by z |--> f is surjective, show that X...

Homework Statement Let \beta be a basis for a finite dimensional inner-product space. b) Prove that is < x, z > = < y, z> for all z \in \beta, then x = y Homework Equations The Attempt at a Solution start with the Cauchy-Schwarz: |< x, z >| \leq ||x|| ||z|| then because <x,z>...

An inner product space can be both normal and self adjoint, correct?

Let M_2x2 denote the space of 2x2 matrices with real coeffcients. Show that (a1 b1) . (a2 b2) (c1 d1) (c2 d2) = a1a2 + 2b1b2 + c1c2 + 2d1d2 defines an inner product on M_2x2. Find an orthogonal basis of the subspace S = (a b) such that a + 3b - c = 0 (c d) of M_2x2...

Homework Statement Define an inner product on P2 by <f,g> = integral from 0 to 1 of f(x)g(x)dx. find an orthonormal basis of P2 with respect to this inner product. Homework Equations So this is a practice problem and it gives me the answer I just don't understand where it came from...

Homework Statement Given g\equiv g_{ij} = [-1 0; 0 1] Show that A= A^{i}_{j} = [1 2 -2 1] is symmetric wrt innter product g, has complex eigenvalues, but eigenvectros have zero length wrt the complex inner product. The Attempt at a Solution Im sure this is just a simple...

I am having difficulties with understanding some aspects of inner products. For example, ||u||² = <u,u> Where <u,u> denoted the inner product of "u" with itself. My problem here is that we can define any inner product we wish. For example, if I defined, <u,v> = u1v1 + 3(u2v2) Then...

Homework Statement State why (u,v) is not an inner product for u=(u1,u2) and v=(v1,v2) in R2 (u,v)=-u2v2 Homework Equations (u,v)=(v,u) c(u,v)=(cu,v) (v,v)=>0 and (v,v)=0 if only if v=0 The Attempt at a Solution I am having trouble understanding this problem and how to start it...

Homework Statement Let V be a vector space over a field F = R or C. Let W be an inner product space over F. w/ inner product <*,*>. If T: V->W is linear, prove <x,y>' = <T(x),T(y)> defines an inner product on V if and only if T is one-to-one Homework Equations What we know, W is an inner...

Last two inner product questions. The first one I am little confused on and the second one I don't know what to do. See worked attached.

I have attached the solutions of parts a, b, and what I have done for part c. My part c isn't going to turn out correct and I don't know what is wrong.

In C[0,1], with inner product defined by (3), consider the vectors 1 and x. Find the angle theta between 1 and x. (3)\int_{0}^{1}f(x)g(x)dx Find the angle theta between 1 and x I don't know what to do with polynomial inner product vector space

Hello, I thought I understood the Dot Product but Apparently Not! \overline{u} \ \cdot \ \overline{v} \ = (u_x \ \cdot \ v_x) ( \overline{i} \cdot \overline{i} ) \ + \ (u_y \ \cdot \ v_y) ( \overline{j} \cdot \overline{j} ) \ = \ | \overline{u} | | \overline{v} | cos \theta That is the...

Is the first part of this question saying find a scalar a such that \int_{-1}^1 \! a^{2} \, dx \, =1 \,? In that case I believe 1/20.5 is an answer...or am I reading the notation wrong? Thanks.

this question is in reference to eq 3.9 and footnote 6 in griffith's intro to quantum mechanics consider a function f(x). the inner product <f|f> = int [ |f(x)|^2 dx] which is zero only* when f(x) = 0 only points to footnote 6, where Griffith points out: "what about a function that is...

Homework Statement Show that \psi (\gamma^a\phi)=-(\gamma^a\phi)\psi Homework Equations Maybe \{\gamma^a, \gamma^b\}=\gamma^a\gamma^b+\gamma^b\gamma^a=2\eta^{ab}I Perhaps also: (\gamma^0)^{\dag}=\gamma^0 and (\gamma^i)^{\dag}=-(\gamma^i) The Attempt at a Solution The gammas are...

A basic qn:An infinite sum of vectors will also be a vector in the same vector space? By definition, the sum of any two vectors of a vector space will be a vector in the same vector space. But does this mean the sum of an uncountable or countable number of vectors will also be a vector in the...

Hi, I'm having trouble understanding how to perform the following calculation: u=(u,v,w) (\nabla u + (\nabla u)^T) : \nabla u I get the following by doing the dot product of the first term and then adding the dot product of the second term, but I'm pretty...

Homework Statement We are given a linear map f, f:R2xR2->R. f has the following properties: 1)It is linear for the changes of the first variable 2)It is linear for the changes of the second variable 3)f((3,8),(3,8))=13 We are asked to say if f is anyhow related to the normal inner...

I was reading "Principles of Quantum Mechanics" - Shankar, and I'm having trouble understanding the inner product. Can someone help me or link me to a site that explains it? The axioms of the inner product are 1. \langle V|W\rangle = \langle W|V\rangle^* 2. \langle V|V\rangle \geq 0\ \...

Hey guys, In one of the questions for our assignment we have to decide whether <v,w> = v^{}TAw (with a conjugate bar over w) defines an inner product on C3. We are given three 3x3 marices to test this. What is the procedure for doing this? Do we just give w and v values such as a1, a2, a3...

once i solve that one is the derivative of the other but here its much harder to guess the formulla http://i47.tinypic.com/ixt74i.jpg what is the general method?

I am trying to understand the definition of the inner product of two functions on an interval. I know that the form of a scalar product in finite dimensional space is given by \vec{\phi} \bullet \vec{\psi}= \sum_{k} \phi_{k} \psi_{k} and in infinite dimensional space \langle \phi , \psi...

Homework Statement Define the inner product of two polynomials, f(x) and g(x) to be < f | g > = ∫-11 dx f(x) g(x) Let f(x) = 3 - x +4 x2. Determine the inner products, < f | f1 >, < f | f2 > and < f | f3 >, where f1(x) = 1/2 , f2(x) = 3x/2 and f3(x) = 5(1 - 3 x2)/4 Expressed as a...

Let V be an inner product space, and let W be a finite-dimensional subspace of V. If x\notin W, prove that there exists y\in V such that y \in W(perp), but <x,y>\neq 0. I don't have a clue... Thanks

Homework Statement We consider P2 the vector space of all real polynomials of degree at most 2. Show that <f,g> = \int_{-1}^{1}f(x)g(x)dx defines an inner product space Homework Equations I'm Using one of the Axioms of Inner product spaces IP1. which states that. <u,u>...

Hi, I was wondering how would i determine if <p,q> = p(0)q(0)+ p(1)q(1) is an inner product for P2. I know, we have to check for non-negativity, symmetry and linearity. Just not sure how. thanks!

Homework Statement Show that <U,V> = u1.v1 + u2.v3 + u2.v3 + u4.v4 is NOT an inner product on M[SIZE="2"]2x2Homework Equations U: row 1 = [u1 u2] row 2 = [u3 u4] V: row 1 = [v1 v2] row 2 = [v3 v4] The Attempt at a Solution As I went through each of the axioms, I found that they were all...

Homework Statement Let v,w be vectors in a complex inner product space such that ||v|| = 1, ||w|| = 3 and <v,w> = 1 + 2i. Find ||v + iw||. Homework Equations The properties of an inner product. The Attempt at a Solution I figured ||v+iw||^2 = <v+iw,v+iw> Then using the...

Homework Statement I have two assignments I have some problems with. The first one: For n > 0, let fn(t) = {1, 0 \leq t \leq 1/n 0, otherwise Show that fn \rightarrow 0 in L2[0,1]. Show that fn does NOT converge to zero uniformly on [0,1] The second one: Find...

Hi I am kinda new to this topic two . I was wondering how can I prove that the following expressions define scalar product. All I can guess that I need to show that they follow the properties of the scalar product. But how? If possible, help me with an example .1. (f,g)=\int f(x)g(x)w(x)dx...

What is the relationship btw the Hermitian inner product btw 2 complex vectors & angle btw them. x,y are 2 complex vectors. \theta angle btw them what is the relation btw x^{H}y and cos(\theta)?? Any help will be good?

Homework Statement http://img199.imageshack.us/img199/3230/mathquestion2.png Thank you very much in advance! EDIT: [SIZE="3"]subspace U = span(1,t)

Homework Statement Suppose V is a real inner-product space and (v1, . . . , vm) is a linearly independent list of vectors in V. Prove that there exist exactly 2^m orthonormal lists (e1, . . . , em) of vectors in V such that span(v1, . . . , vj) = span(e1, . . . , ej) for all j ∈ {1, . . . ...

Homework Statement Prove that (\sumajbj)2\leq\sumjaj2*\sum(bj)2/j with j from 1 to n. for all real numbers a1...an and b1...bn Homework Equations The Attempt at a Solution I can prove this using algebra, but how is it done using inner product concepts? If someone could start me up...

Hello, Can someone help me understand why the Inner Product of a Null vector with itself can be non zero if complex numbers are involved? And why using the complex conjugate resolved this? I may have understood this wrong. It could be that an Inner Product of any non-Null vector with...

Hi. I've found the following relation (in a book about the qm 3-body scattering theory): <\Omega^{\pm}^{\dagger} \Psi_n|p>= ... = 0 where |p> is a momentum eigenstate. So it is shown, that the inner Product is zero. Then they conclude that \Omega^{\pm}^{\dagger}|\Psi_n> = 0 because the...

Homework Statement For each of the following inner product spaces V (over F) and linear transformation g:=V \rightarrow F, find a vector y such that g(x) = <x,y> for all x element of V. The particular case I'm having trouble with is: V=P2(R), with <f,h>=\int_0^{1} f(t)h(t)dt ...

-------------------------------------------------------------------------------- I understand the concepts of the inner product in Rn as well as the vector space of C[a,b] as the integral operator, however i don't understand how to obtain or prove the inner product space of two 2x2 matrices...