Inner product Definition and 312 Threads

Homework Statement Find the inner product of f(x) = σ(x-x0) and g(x) = cos(x) Homework Equations ∫f(x)*g(x)dx Limits of integration are -∞ to ∞ The Attempt at a Solution First of all, what is the complex conjugate of σ(x-x0)? Is it just σ(x-x0)? And I'm not sure how to...

i realize this is a linear algebra question, but the bra-ket notation is still a little confusing to me so i posted it in this section. |e>=(1+i,1,i) (n-tuple representation, where i's are the imaginaries) so the norm of this would then be the following...

Homework Statement Given ##S = \{1, x, x^2\}##, find the coordinates of ##x^2 + x + 1## with respect to the orthogonal set of S.Homework Equations Inner product on polynomial space: ##<f,g> = \int_{0}^{1} fg \textrm{ } dx## The Attempt at a Solution I used Gram-Schmidt to make ##S## orthogonal...

Hi, I know these questions must sound ridiculous and I apologize, I'm a newbie. My textbook says that the inner product of the momentum four-vector is P\bulletP=P\bulletP - E^{2}/c^{2}=-m^{2}*c^{2} So my silly questions are: 1) where did the - E^{2}/c^{2} term come from? 2) I know I'm being...

I was wondering about the proper way to say, \langleA|B\rangle . I have recently heard, "The inner product of A with B." But I'm not sure if this is correct. Does anyone know the proper order in which to place A and B in the sentence? As a simple example: Suppose you're speaking with...

Correct me if I'm wrong here but it is my understanding that vector spaces are given structure such as inner products, because it allows us to use these structured vector spaces to describe and analyse physical things with them. So physical properties such as 'distance' cannot be analysed in...

Let A be a Hermitian operator with n eigenkets: A|u_i\rangle = a_i |u_i\rangle for i=1,2,...,n. Suppose B is an operator that commutes with A. How could I show that \langle u_i | B | u_j \rangle = 0 \qquad (a_i \neq a_j)? I have tried the following but not sure how to proceed: AB -...

So, I have an equivalence I need to prove, but I think I'm having trouble understanding the problem at a basic level. The problem is to prove that the inner product of a and b equals 1/4[|a+b|^2 - |a-b|^2] (a, b in C^n or an n-dimensional vector space with complex elements). I don't...

Let V be a real vector space. Suppose to each pair of vectors u,v ε v there is assigned a real number, denoted by <u, v>. This function is called a inner product on V if it satisfies some axioms. 1. What does refers by "this function"? Is it "<u, v>"? If it is then How we can call it's a...

Consider 2 atomic orbitals with wave function a: σ(r), b: μ(r) in a diatomic molecules. σ(r) (or μ(r)) is localized around an atom a (or b) and is relevant for the discussion of the molecular orbital. These orbitals are orthogonal and normalised. The creation operators are x, y and vacuum, |0>...

When one says that <\varphi|\psi> is the probability that \psi collapses to \varphi, does this "collapse" necessarily involve a measurement (so that one would have to find the implicit Hamiltonian)? Or does this just exist as part of the evolution of the wave function, perhaps the vacuum energy...

Hello, some time ago I read that if we know the metric tensor g_{ij} associated with a change of coordinates \phi, it is possible to calculate the (Euclidean?) inner product in a way that is invariant to the parametrization. Essentially the inner product was defined in terms of the metric...

Hello, let's assume we have an admissible change of coordinates \phi:U\rightarrow \mathbb{R}^n. I would like to know how the inner product on ℝn changes under this transformation. In other words, what is \left\langle \phi (u), \phi (v) \right\rangle for some u,v \in U ? I thought that...

Homework Statement let \ell^{2} denote the space of sequences of real numbers \left\{a_{n}\right\}^{\infty}_{1} such that \sum_{1 \leq n < \infty } a_{n}^{2} < \infty a) Verify that \left\langle \left\{a_{n}\right\}^{\infty}_{1}, \left\{b_{n}\right\}^{\infty}_{1} \right\rangle =...

Let the inner product on P2(C) be defined as: ∫ (from -1 to 1) p(t)(conjugate) q(t) dt. using the gram schmidt process and the standard basis {1, x, x2} find an orthonormal basis. So my only issue really is interpreting this integral. I just wanted to test if the vectors in the...

Homework Statement For f and g in F[a,b], we define an inner product on F[a,b] by <f,g> = ∫^{b}_{a} f(x)g(x)dx a) Find the angle between the functions f(t) = 5t - 3 and g(t) = t^{3} - t^{2} in F[0,1]. b) Find an orthonormal basis for the subspace of F[0,1] spanned by {1, e^{-x}...

Homework Statement First I'd like to state the meaning of my notations x = (x0,x1,x2...xn) y = (y0,y1,y2...yn) |x| = absolute value of x ||x|| = Normal of x <x,y> = Inner Product of x and y I have to prove the following |<x1,y1> - <x2,y2>| ≤ ||x1 - x2||*||y1|| +...

1) Using the indentity:||u+v||^2=||u||^2+||v||^2+(u,v)+(v,u) - where (x,y) denotes the inner product . Now if we let the second term = iv, my book gives the identity as then: ||u+iv||^2=||u||^2+||v||^2-i(u,v)+i(v,u) [1]. And I am struggling to derive this myself, here is my working...

The question is : If the vector space C[1,1] of continuous real valued functions on the interval [1,1] is equipped with the inner product defined by (f,g)=^{1}_{-1} \intf(x)g(x)dx Find the linear polynomial g(t) nearest to f(t) = e^t? So I understand the solution will be given by...

I am attempting to work my way through the product rule for inner products, using the properties of linearity and symmetry. I am wondering if the following step is allowed, exploiting the bilinear property: f(t) = \left\langle{\alpha}(t),{\beta}(t)\right\rangle \rightarrow f'(t) = \lim_{h \to...

My analysis instructor has posed an exercise to me in the following format: "For {\bf{u}} = (u_1,u_2), {\bf{v}} = (v_1,v_2) \in R^2 define \left\langle{\bf{u}},{\bf{v}}\right\rangle = 3u_1u_2 - u_1v_2 - u_2v_1 + \frac{1}{2}u_2v_2.. Show that this is an inner product on ##R^2##." Not sure what...

I am working on this problem, and I am having difficulties with a certain part of a proof: If A is a C^*-algebra. And X is a Hilbert A-module. Can we say that \langle X,X \rangle has an approximate identity e_\alpha = \langle u_\alpha , v_\alpha \rangle such that u_\alpha , v_\alpha are...

let E be an inner product space and (e_n) an orthonormal sequence in E. For x in E and any positive integer n, prove that Re(<x,(<x,e_1>+...<x,e_k>)e_n>)= |<x,e_1>|^2+...+|<x,e_n>|^2 I got <x,(<x,e_1>+...<x,e_k>)e_n>= <<x,e_1>e_1,x>+...<<x,e_n>e_n,x> but haven't a clue how to find the real...

consider C[0,2], the set of continuous functions from [0,2] to C. The inner product is <f,g> = the integral of f(t)g(t)* from 0 to 2. show that: sqrt(2)||f|| is greater than or equal to the magnitude of the integral of f from 0 to 2, where ||.|| is the norm of f.

I'm okay on proving the other properties, just struggling with what to do on this one: (v,v)≥0, with equality iff v=0,where the inner product is defined as: z1w*1+iz2w*2 (where * represent the complex conjugate) My working so far is: u1u*1+iu2u*2 =u1^2 + iu2^2 (I'm not sure what to do...

Sure, this could also appropriately be placed under linear alg. this being said, can anyone give me an intuitive explanation for the real inner product? i realize it as: <f(x),g(x)>\doteqdot\int_a^bf(x)g(x)dx where i can think of this as an "infinite" dot product along a to b...

Hi I am going through Sheldon Axler - Linear Algebra Done right. The book States the Complex Spectral Theorem as : Suppose that V is a complex inner product space and T is in L(V,V). Then V has an orthonormal basis consisting of eigen vectors of T if and only if T is normal. The proof of this...

The book I am going through says this : The below proposition is false for real inner product spaces. As an example, consider the operator T in R^2 that is a counter clockwise rotation of 90 degrees around the origin. Thus , T(x,y) = (-y,x). Obviously, Tv is orthogonal to v for every v in...

This may be a very silly question, but still apologies, I read in Sheldon Axler, that the inner product of two orthogonal vectors is DEFINED to be 0. Let u,v belong to C^n. I am unable to find a direction of proof which proves that for an nth dimension vector space, if u perp. to v, then <u,v>...

I have three (N x 1) complex vectors, a, b and c. I know the following conditions: (1) a and b are orthonormal (but length of c is unknown) (2) c lies in the same 2D plane as a and b (3) aHc = x (purely real, known) (4) bHc = iy (purely imaginary, unknown) where (.)H denotes...

Homework Statement Let V be a complex inner product space and let S be a subset of V. Suppose that v in V is a vector for which <s,v > + <v,s> \leq <s,s> Prove that v is in the orthogonal set S\bot Homework Equations We have the three inner product relations: 1) conjugate symmetry...

Let x = (2,1+i,i) and y = (2-i,2,1+2i). Find <x,y> So my work is the following: 2(2-i) + (1+i)2 + i(1+2i) = 4+i, but my book says the correct answer is 8+5i. Hmmm what am I doing wrong?

I been reading some material that lead me to understand that it takes an inner product of a dyad and a vector to obtain another vector at an angle to the initial one... cross product among two vectors would be an option only if we are willing to settle to a right angle. After few days i...

Hi, if we suppose x and y are two elements of some vector space V (say ℝn), and if we consider a linear function f:V→V', we know that the inner product of the transformed vectors is given by: \left\langle f\mathbf{x} , f\mathbf{y} \right\rangle = \left\langle \mathbf{x} ...

Homework Statement Let V be the inner product space. Show that if w is orthogonal to each of the vectors u1, u2,...,ur, then it is orthogonal to every vector in the span{u1,u2,...,ur}. Homework Equations u.v=0 to be orthogonal If u and v are vectors in an inner product space, then...

Homework Statement Let V be a real inner product space, and let v1, v2, ... , vk be a set of orthonormal vectors. Prove Ʃ (from j=1 to k)|<x,vj><y,vj>| ≤ ||x|| ||y|| When is there equality? Homework Equations The Attempt at a Solution I've tried using the two inequalities given to us in...

Hey, here's a simple question. I have been reading some materials and, for the n-th time in my life, there was a definition of an inner product as a function V \times V \rightarrow F, where V is an abstract vector space and F is an underlying scalar field. However, it got me thinking. Inner...

Consider P5(R) together with innner product < p ,q > = ∫p(x)q(x) dx. Find W-perp if W = {p(x) \in P5(R) : p(0) = p'(0) = p''(0) = 0} Attempt: I am having trouble with the condition. I always have trouble with these conditions. SO as of now I am going to let q(x) be the standard basis of...

I would like if someone could either verify or clarify my thinking about inner products. There is a matrix, V that is m x n, that is made up of complex numbers. When matrix V is multiplied by its hermitian then the product is a matrix with the same integer down the main diagonal (i.e...

Homework Statement Let V be an inner product space. For v ∈ V fixed, show that T(u) =< v, u > is a linear operator on V . Homework Equations The Attempt at a Solution First to show it is a linear operator, you show that T(u+g)=T(u)+T(g) and T(ku)=kT(u) So, T(u+g)=<v...

Homework Statement If x = (x1, x2) and y = (y1, y2)... Show that <x,y> = 3(x1)(y1) - (x1)(y2) - (x2)(y1) + 3(x2)(y2) Homework Equations I know that to define it as an inner product space, the following must be correct: <x,y> = <y,x> a<x,y> = <ax,y> <x,y+z> = <x,y> + <x,z>...

Just starting up school again and having trouble remembering some mathematics. Here's the problem. Find the inner product of ⃗a = (1, 45◦) and ⃗b = (2, 90◦), where these vectors are in polar coordinates (r, θ). Thanks =) 1st post here btw.

Hi, Let H^2 be the Hardy space on the open unit disk. I am wondering how can I compute the following inner product <\frac{1}{\left(1-\overline{\alpha_1} z\right)^2}\frac{z-\alpha_2}{1-\overline{\alpha_2} z},\frac{z}{\left(1-\overline{\alpha_1} z\right)^2}>, where \alpha_1,\alpha_2 in the unit...

Homework Statement prove the triangle inequality ||v+w|| ≤ ||v||+||w|| if (ℂ, V, + , [ , ] ) is a Hermitic space where [ , ] is the Hermitic product. Homework Equations |[v,w]| ≤ ||v|| . ||w|| \overline{a+b i}= a-b i (where i is the imaginary number, a+bi a complex number)...

In inner product spaces of polynomials, what is the point of finding the angle and distance between two polynomials? How does the distance and angle relate back to the polynomial?

In inner product spaces of polynomials, what is the point of finding the angle and distance between two polynomials? How does the distance and angle relate back to the polynomial?

What is the most motivating way to introduce general inner product spaces? I am looking for examples which have a real impact. For Euclidean spaces we relate the dot product to the angle between the vectors which most people find tangible. How can we extend this idea to the inner product of...

What is the most motivating way to introduce general inner product spaces? I am looking for examples which have a real impact. For Euclidean spaces we relate the dot product to the angle between the vectors which most people find tangible. How can we extend this idea to the inner product of...

The isomorphism of ℝ5 and P4 is obvious for the "standard" inner product space. The following question arise from an example in my course literature for a course in linear algebra. The example itself is not very difficult, but there is a statement without any proof, that if the inner product...

Hello all. I have a fairly rudimentary knowledge of matrices and broader linear algebra. This gets me in a lot of trouble when I'm following along the math of something fine and then I run into some matrix stuff and get stumped, like this. I'm a little bit confused on taking the inner product...