Inner product Definition and 312 Threads

Hey all, I've never taken a formal class on tensor analysis, but I've been trying to learn a few things about it. I was looking at the metric tensor in curvilinear coordinates. This Wikipedia article claims that you can formulate a dot product in curvilinear coordinates through the following...

Let x be in R^n and Q in Mat(R,n) where Q is hermitian and negative definite. Let (.,.) be the usual euclidian inner product. I need to prove the following inequality: (x,Qx) <= a(x,x) where "a" is the maximum eigenvalue of Q. Any idea?

Homework Statement Show that the sup norm on R2 is not derived from an inner product on R2. Hint: suppose <x,y> is an inner product on R2 (not the dot product) and has the property that |x|=<x,y>0.5. Compute <x±y, x±y> and apply to the case x=e1, y=e2.Homework Equations |x|=<x,y>0.5 I've...

Homework Statement Consider the set ##C^2= {x=(x_1,x_2):x_1,x_2 \in C}##. Prove that ##<x,y>=x_1 \overline{y_1}+x_2 \overline{y_2}## defines an inner product on ##C^2## Homework Equations The Attempt at a Solution ##<,y>=\overline {<y,x>}## ##= \overline {y_1x_1} +...

Homework Statement An inner product is linear in both components. Homework Equations <x,y> = <conjugate(y),conjugate(x)> <x+y,z> = <x,z> +<y,z> The Attempt at a Solution I thought it was true . It is obvious that it is linear for the first component by definition Attempt to show...

Weinberg in his 1st book on QFT writes in the paragraph containing 2.5.12 that we may choose the states with standard momentum to be orthonormal. Isn't that just true because the states with any momentum are chosen to be orthonormal by the usual orthonormalization process of quantum mechanics...

Hello buddies, Here is my question. It seems simple but at the same time does not seem to have an obvious answer to me. Given that you have two vectors \mathbf{u},\mathbf{v}. They are orthogonal \mathbf{u}^T\mathbf{v}=0 by standard dot product definition. They have norm one...

i m really confused..please explain with a physical example so that I can learn the application of it

Homework Statement Show that the inner product of the Pauli matrices, σ, and the momentum operator, \vec{p}, is given by: σ \cdot \vec{p} = \frac{1}{r^{2}} (σ \cdot \vec{r} )(\frac{\hbar}{i} r \frac{\partial}{\partial r} + iσ \cdot \vec{L}), where \vec{L} is the angular momentum operator and...

Homework Statement P5 is an inner product space with an inner product. We applied the Gram Schmidt process to the basis {1,x,x^2,x^3,x^4} and obtained the following result. {f1,f2,f3,f4,x^4+2} What is the orthogonal complement of P3 in P5 with respect to this inner product?Homework Equations...

On a finite-dimensional vector space over R or C, is every norm induced by an inner product? I know that this can fail for infinite-dimensional vector spaces. It just struck me that we never made a distinction between normed vector spaces and inner product spaces in my linear algebra course...

I have just realized that I accidently put it in wrong sub forum. This should be in 'calculus and beyond'. Homework Statement Prove the function <x,y>=x_1y_1+x_2y_2+x_3y_3 defines an inner product space on the real vector space R^3 where x=(x1,x2,x3) and y=(y1,y2,y3)The Attempt at a Solution...

Show the taxicab norm is not an IP. taxicab norm is v=(x_{1}...x_{n}) then ||V||= |x_{1}|+...+|x_{n}|) I was thinking about using the parallelogram law but I would get this nasty...

Sometimes the way authors write a book makes you wonder if they are at their wits' end: 'A vector space with an inner product is an inner product space!' I am not sure if I have gone crazy but to me it is obvious that if you have a vector space wrapped up in a nice gift basket and sent off to...

Hi, I have a bilinear form defined as g : ℝnxℝn->ℝ by g(v,w) = v1w1 + v2w2 + ... + vn-1wn-1 - vnwn I have to show that g is an inner product, so I checked that is bilinear and symmetric, but how to show that it's nondegenerate too?

I typed the problem in latex and will add comments below each image. The supremum of |1 - x| seems dependent on the interval [a, b]. For instance, if [a, b] = [-500, 1], then 501 is the supremum of |1 - x|. But if [a, b] = [-1, 500], then 499 is the supremum of [1 - x]. So what should I...

Homework Statement Let u = (u1, u2, u3) and v = (v1, v2, v3) Determine if it's an inner product on R3. If it's not, list the axiom that do not hold. Homework Equations the 4 axioms to determine if it's an inner product are (all letters representing vectors) 1. <u,v> = <v,u> 2...

Consider the maximum inner product of a density operator with a maximally entangled state. (So, given a density operator, we're maximizing over all maximally entangled states.) I'm pretty sure the minimum value (a lower bound on the maximum) for this is 1/n2, using the density operator 1/n2...

Why is the inner product of two orthogonal vectors always zero? For example, in the real vector space R^n, the inner product is defined as ||a|| * ||b|| * cos(theta), and if they are orthogonal, cos(theta) is zero. I can understand that, but how does this extend to any euclidean space?

Friedberg's Linear Algebra states in one of the exercises that an inner product space must be over the field of real or complex numbers. After looking at the definition for while, I am still having trouble seeing why this must be so. The definition of a inner product space is given as follows...

Homework Statement Assume the inner product is the standard inner product over the complexes. Let W= Spanhttp://img151.imageshack.us/img151/6804/screenshot20111122at332.png Find an orthonormal basis for each of W and Wperp.. The Attempt at a Solution Obviously I need to use Gram-Schmidt...

I know the cross product and dot product of euclidean space R^3. But I wanted to know if there is a way of thinking the cross product "in terms of" the dot product. That is because the dot product can be generalized to an inner product, and from R^3 to an arbitrary inner vector space (and...

Space of continuous functions. Inner product <f,g>=\int_{-1}^{1}f(x)g(x)dx. Find a monic polynomial orthogonal to all polynomials of lower degrees. Taking a polynomial of degree 3. x^3+ax^2+bx+c Need to check \gamma, x+\alpha, x^2+\beta x+ \lambda \int_{-1}^{1}(\gamma...

Homework Statement C[a,b] is a vector space of continuos real valued functions. for f,g in C[a,b] <f,g>=∫f(x)g(x)dx, [a,b] Give a completely rigorous proof that if <f,f>=0, then f=0 2. The attempt at a solution I tried to prove this by contrapositive, "f≠0 implies that <f,f>≠0 When...

Homework Statement Is the following an inner product space if the functions are real and their derivatives are continuous: <f(t)|g(t)> = \int_0^1 f'(t)g'(t) + f(0)g(0) Homework Equations I was able to prove that it does satisfy the first 3 conditions of linearity and that...

Inner product: \displaystyle <f,g>=\frac{1}{\pi}\int_{-\pi}^{\pi}fg \ dx=\begin{cases}0 & \ \text{if} \ f=g\\1 & \ \text{if} \ f\neq g\end{cases} Basis: \displaystyle\left\{\frac{1}{\sqrt{2}},\cos\theta, \sin\theta,\cdots\right\} I am trying to remember how to integrals of the form...

I'm struggling with the meaning of inner products between vectors and covectors. Consider dual space V^{\ast} of an inner product space V over field K, with linear forms \vec{v}\:^{\ast}\in V^{\ast} \vec{v}\:^{\ast}:V\rightarrow K:\vec{w}\rightarrow\left<\vec{v},\vec{w}\right> In...

When studying complex numbers/vectors/functions and so forth you constantly encounter the idea of an inner product of two quantities (numbers/vectors/functions). It's represented as A*B is the inner product of two of these, but I've never been convinced why it couldn't also be AB*, as this is...

Homework Statement \int d^{3} \vec{r} ψ_{1} \hat{A} ψ_{2} = \int d^{3} \vec{r} ψ_{2} \hat{A}* ψ_{1} Hermitian operator A, show that this condition is equivalent to requiring <v|\hat{A}u> = < \hat{A}v|u> Homework Equations I changed the definitions of ψ into their bra-ket forms...

Using the standard inner product in ℂ^n how would I calculate the norm of: V= ( 1 , i ) , where this is a 1 x 2 matrix

Hi everyone! I would like to ask how would you verify if functions form an inner product space? For example, if one has functions of the form q(x)e^-(x^2/2) where q(x) is a polynomial of degree < N in x, on the interval -∞ < x < ∞. Also, how would you specify the dimension of the space, if it...

Homework Statement Consider the linear space S, which consists of square integrable continuous functions in [0,1]. These are continuous functions x : [0,1] -> R such that the integral is less than infinity. Homework Equations Show that the operation ∫x(t)y(y)dt at [0,1] is an inner product...

Homework Statement Let R4 have the Euclidean inner product. Find two unit vectors that are orthogonal to the three vectors u = (2, 1, -4, 0) ; v = (-1, -1, 2, 2) ; w = (3, 2, 5, 4) Homework Equations <u, v> = u1v1 + u2v2 + u3v3 + u4v4 = 0 {orthogonal} The Attempt at a Solution...

Homework Statement Find d(u, v), where the inner product is defined by the matrix [1 2] [-1 3] and u = (-1, 2), v = (2, 5) Homework Equations <u, v> = Au . Av d(u, v) = abs(u - v) The Attempt at a Solution I first tried to find the resulting inner product from the...

Hams and Raedt gave a quantum computational algorithm to calculate the density of states of a spin system. Starting with an initial random state, they obtain the time evolution of the state. Later they take the inner product of the evolved state with the initial state. How does the inner...

Homework Statement Hi suppose you're given two signals for example x_{1}(t) = cos(3 \omega_{0} t) x_{2}(t) = cos(7 \omega_{0} t) and you want to find out the inner product Homework Equations The Attempt at a Solution I mean, it's an integral right? But what will the boundaries be...

In one of my tutorial problems, I was asked to verify if the following function is a valid inner product <x,y>= x1x2 + y1y2 Note, x=(x1,x2)^{T} and y=(y1,y2)^{T} where T means transpose of the matrix The tutor said to us the answer is no because it fails the linearity test Does...

Homework Statement Denote the inner product of f,g \in H by <f,g> \in R where H is some(real-valued) vector space a) Explain linearity of the inner product with respect to f,g. Define orthogonality. b) Let f(x) and g(x) be 2 real-valued vector functions on [0,1]. Could the inner product be...

I am trying to take the derivative of an inner product (in the most general sense over L^2), and was curious if the derivative follows the "chain rule" for inner products. i.e. Does D_y(<f,g>) = <D_y(f),g> + <f,D_y(g)> where D_y is the partial derivative w.r.t. y. So for example...

I have read that the electric and magnetic fields are always "perpendicular". Is that true? And if so, does that mean the inner product of the two vectors is zero? E_x * B_x + E_y * B_y + E_z * B_z = 0 ? Also, is there any special meaning in electrodynamics to the quantity: | B |^2...

[-1]int[1]P(x)Q(x)dx P,Q\inS verify that this is an inner product.

Hello everyone! I am doing some work involves proving that a matrix A (A=U(V^T)) is diagonalizable. In this situation, U and V are non zero vectors in R^n. So far, I have proven that U is an eigenvector for A, and that the corresponding eigenvalue is the inner product of U and V ((U^T)V)...

if A \in C nxn,show that (x,Ay)=0 for all x,y \in C[n], then A=0 (x,Ay) denote standard inner product on C[n]

Homework Statement Given that the vectors \underset{A}{\rightarrow} and \underset{B}{\rightarrow} are parallel transported along a curve: \triangledown _{\underset{l}{\rightarrow}}A = 0 \triangledown _{\underset{l}{\rightarrow}}B = 0 Show that g(A,B) = constant along the curve Homework...

Could someone explain me the difference between the inner product and the dot product? Thanks all

Homework Statement Hello everyone! I am reading abour Poincares duality between 2 cohomology groups and here comes an inner product defined as \left\langle \omega, \eta\right\rangle \equiv \int_{M}\omega \wedge \eta and then the author of my book says ''The product is bilinear and...

Homework Statement Prove that any non-degenerate inner product on \mathbb{R}^{n} is, as a non-degenerate inner product space, isomorphic to \mathbb{R}^{p, n-p} for some 0 \le p \le n. Homework Equations The non-degenerate inner product on \mathbb{R}^{p, n-p} is defined as...

Hello all, I am having trouble showing that the operation defined by f*g(f of g)= Integral[from a to b]f(x)g(x) is an inner product. I know it must fulfill the inner product properties, which are: x*x>=0 for all x in V x*x=0 iff x=0 x*y=y*x for all x,y in V x(y+z)=x*z+y*z...

Homework Statement Suppose T is in the set of linear transformations on an inner product space V and lamba is in F, the scalar field of V. Prove that lamda is an eigenvalue of T iff lamda_bar (the conjugate) is an eigenvalue of T*, the adjoint of T. Homework Equations Using []...

What is the motivation behind defining the inner product for a vector space over a complex field as <v,u> = v1*u1 + v2*u2 + v3*u3 where * means complex conjugate as opposed to just <v,u> = v1u1 + v2u2 + v3u3 They both give you back a scalar. The only reason I can see is the special case for...