# What is Orthonormal basis: Definition and 68 Discussions

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for Rn arises in this fashion.
For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of Rn under dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process.
In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces. Given a pre-Hilbert space H, an orthonormal basis for H is an orthonormal set of vectors with the property that every vector in H can be written as an infinite linear combination of the vectors in the basis. In this case, the orthonormal basis is sometimes called a Hilbert basis for H. Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required. Specifically, the linear span of the basis must be dense in H, but it may not be the entire space.
If we go on to Hilbert spaces, a non-orthonormal set of vectors having the same linear span as an orthonormal basis may not be a basis at all. For instance, any square-integrable function on the interval [−1, 1] can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonormal basis), but not necessarily as an infinite sum of the monomials xn.

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1. ### I Gram Schmidt to make unitary

I understand the rationale for using the Gram-Schmidt process to find an orthogonal (or orthonormal) basis from a given set of linearly independent vectors (e.g., eigenvectors of a Hermitian matrix). However, the rational for using it on the columns of a matrix in order to get a unitary matrix...
2. ### I cannot make the Gram-Schmidt procedure work

This is problem A.4 from Quantum Mechanics – by Griffiths & Schroeter. I cannot make the Gram-Schmidt procedure work. I don't know whether I am just inept with complex vectors or I have made some wrong assumption. The Gram-Schmidt procedure (modified, I think) Suppose you start with a basis...
3. ### I Orthonormal basis expression for ordinary contraction of a tensor

I'm reading Semi-Riemannian Geometry by Stephen Newman and came across this theorem: For context, ##\mathcal{R}_s:Mult(V^s,V)\to\mathcal{T}^1_s## is the representation map, which acts like this: $$\mathcal{R}_s(\Psi)(\eta,v_1,\ldots,v_s)=\eta(\Psi(v_1,\ldots,v_s))$$ I don't understand the...
4. ### I Proof that if T is Hermitian, eigenvectors form an orthonormal basis

Actual statement: Proof (of Mr. Tom Apostol): We will do the proof by induction on ##n##. Base Case: n=1. When ##n=1##, the matrix of T will be have just one value and therefore, the characteristic polynomial ##det(\lambda I -A)=0## will have only one solution. So, the Eigenvector...
5. ### MHB Orthonormal basis for the poynomials of degree maximum 2

Hey! 😊 We consider the inner product $$\langle f,g\rangle:=\int_{-1}^1(1-x^2)f(x)g(x)\, dx$$ Calculate an orthonormal basis for the poynomials of degree maximum $2$. I have applied the Gram-Schmidt algorithm as follows: \begin{align*}\tilde{q}_1:=&1 \\...
6. ### MHB Orthonormal basis - Set of all isometries

Hey! 😊 Let $1\leq n\in \mathbb{N}$ and $\mathbb{R}^n$. A basis $B=(b_1, \ldots, b_n)$ of $V$ is an orthonormal basis, if $b_i\cdot b_j=\delta_{ij}$ for all $1\leq i,j,\leq n$. Let $E=(e_1, \ldots,e_n)$ be the standard basis and let $\phi \in O(V)$. ($O(V)$ is the set of all isometries...
7. ### I Orthonormal Basis - Definition & Examples

Is this correct? If not any hints on how to find Many thanks
8. ### I Orthonormal Basis of Wavefunctions in Hilbert Space

Hello, I've a fundamental question that seems to keep myself confused about the mathematics of quantum mechanics. For simplicity sake I'll approach this in the discrete fashion. Consider the countable set of functions of Hilbert space, labeled by i\in \mathbb{N} . This set \left...
9. ### Gram-Schmidt for 1, x, x^2 Must find orthonormal basis

Homework Statement Find orthonormal basis for 1, x, x^2 from -1 to 1. Homework Equations Gram-Schmidt equations The Attempt at a Solution I did the problem. My attempt is attached. Can someone review and explain where I went wrong? It would be much appreciated.
10. ### I Expanding a given vector into another orthonormal basis

Equation 9.2.25 defines the inner product of two vectors in terms of their components in the same basis. In equation 9.2.32, the basis of ## |V \rangle## is not given. ## |1 \rangle ## and ## |2 \rangle ## themselves form basis vectors. Then how can one calculate ## \langle 1| V \rangle ## ? Do...
11. ### Expectation Value of Q in orthonormal basis set Psi

Homework Statement Suppose that { |ψ1>, |ψ2>,...,|ψn>} is an orthonormal basis set and all of the basis vectors are eigenvectors of the operator Q with Q|ψj> = qj|ψj> for all j = 1...n. A particle is in the state |Φ>. Show that for this particle the expectation value of <Q> is ∑j=1nqj |<Φ|...
12. ### MHB Orthonormal Basis times a real Matrix

Hi! I have an orthonormal basis for vector space $V$, $\{u_1, u_2, ..., u_n\}$. If $(v_1, v_2, ..., v_n) = (u_1, u_2, ... u_n)A$ where $A$ is a real $n\times n$ matrix, how do I prove that $(v_1, v_2, ... v_n)$ is an orthonormal basis if and only if $A$ is an orthogonal matrix? Thanks!
13. ### I Orthogonal 3D Basis Functions in Spherical Coordinates

I'd like to expand a 3D scalar function I'm working with, ##f(r,\theta,\phi)##, in an orthogonal spherical 3D basis set. For the angular component I intend to use spherical harmonics, but what should I do for the radial direction? Close to zero, ##f(r)\propto r##, and above a fuzzy threshold...
14. ### Orthonormal basis of 1 forms for the rotating c metric

Homework Statement Write down an orthonormal basis of 1 forms for the rotating C-metric [/B] Use the result to find the corresponding dual basis of vectorsSee attached file for metric and appropriate equations The two equations on the left are for our vectors. the equations on the right...
15. ### Linear Combination Proof of Orthonormal basis

Homework Statement Assume that (|v_1>, |v_2>, |v_3>) is an orthonormal basis for V. Show that any vector in V which is orthogonal to |v_3> can be expressed as a linear combination of |v_1> and |v_2>.Homework Equations Orthonormality conditions: |v_i>*|v_j> = 0 if i≠j OR 1 if i=j. The Attempt...
16. ### Scalar Product of Orthonormal Basis: Equal to 1?

What is the scalar product of orthonormal basis? is it equal to 1 why is a.b=ηαβaαbβ having dissimilar value
17. ### Orthonormal Basis: Definition & Scalar Product in GR

What is an orthonormal basis?? How is the scalar product of orthonormal basis in GR-- a.b = ηαβaαbβ. Please explain
18. ### Finding the orthonormal basis for cosine function

Homework Statement si(t) = √(((2*E)/T)*cos(2*π*fc*t + i*(π/4))) for 0≤t≤T and 0 otherwise. Where i = 1, 2, 3, 4 and fc = nc/T, for some fixed integer nc. What is the dimensionality, N, of the space spanned by this set of signal? Find a set of orthonormal basis functions to represent this set of...
19. ### Euclidean space: dot product and orthonormal basis

Dear All, Here is one of my doubts I encountered after studying many linear algebra books and texts. The Euclidean space is defined by introducing the so-called "standard" dot (or inner product) product in the form: (\boldsymbol{a},\boldsymbol{b}) = \sum \limits_{i} a_i b_i With that one...
20. ### How do you derive relativistic tensors in an orthonormal basis?

I have been recently trying to derive the Einstein tensor and stress energy momentum tensor for a certain traversable wormhole metric. In my multiple attempts at doing so, I used a coordinate basis. My calculations were correct, but the units of some of the elements of the stress energy momentum...
21. ### Finding an orthonormal basis for a subspace

Homework Statement Find an orthonormal basis for the subspace of V4 spanned by the given vectors. x1 = (1, 1, 0, 1) x2 = (1, 0, 2, 1) x3 = (1, 2, -2, 1) Homework Equations Gram-Schmidt Process The Attempt at a Solution I have used the Gram-Schmidt process but seem to be running into trouble...
22. ### Orthonormal Basis: Show A is Self-Adjoint

Hi. Been looking at a question and its solution and I'm confused. Question is - Let ##ψ_n## ,n=1,2,... be an orthonormal basis consisting of eigenstates of a Hamiltonian operator H with non-degenerate eigenvalues ##E_n##. Let A be a linear operator which acts on the energy eigenstates ##ψ_n##...
23. ### Can any vector be in orthonormal basis?

okay so I'm having some conceptual difficulty given some vector space V (assume finite dimension if needed) which has some orthonormal basis i'm given a vector x in V (assume magnitude 1 so it is normalized) now my question is: can x belong to some orthonormal basis of v...
24. ### MHB Extend to an orthonormal basis for R^3

Hello MHB, (I Hope the picture is read able) this is a exemple on My book ( i am supposed to find a singular value decomposition) well My question is in the book when they use gram-Schmidt to extand they use (u_1,u_2,e_3) but I would use (u_1,u_2,e_1) cause it is orthogonal against u_2 which...
25. ### Metric orthonormal basis

Homework Statement For the orthonormal coordinate system (X,Y) the metric is \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} Calculate G' in 2 ways. 1) G'= M^{T}*G*M 2) g\acute{}_{ij} = \overline{a}\acute{}_{i} . \overline{a}\acute{}_{j} Homework Equations \begin{pmatrix}...
26. ### MHB Finding an Orthonormal Basis

Hi everyone, :) Here's a question with my answer. It's pretty simple but I just want to check whether everything is perfect. Thanks in advance. :) Question: Let $$f:\,\mathbb{C}^2\rightarrow\mathbb{C}^2$$ be a linear transformation, $$B=\{(1,0),\, (0,1)\}$$ the standard basis of...
27. ### Hilbert space, orthonormal basis

My book says that "the countability of the ONS in a hilbert space H entails that H can be represented as closure of the span of countably many elements". I must admit my english is probably not that good. At least the above quote does not make sense to me. What is it trying to say? Previously...
28. ### Vector Analysis - Similarities on Orthonormal Basis

Homework Statement Let L: R2 → Rn be a linear mapping. We call L a similarity if L stretches all vectors by the same factor. That is, for some δL, independent of v, |L(v)| = δL * |v| To check that |L(v)| = δL * |v| for all vectors v in principle involves an infinite number of...
29. ### Linear algebra: orthonormal basis

Homework Statement ##\phi## is an endomorphism in ##\mathbb{E}^3## associated to the matrix (1 0 0) (0 2 0) =##M_{\phi}^{B,B}##= (0 0 3) where B is the basis: B=((1,1,0),(1,-1,0),(0,0,-1)) Find an orthonormal basis "C" in ##\mathbb{E}^3## formed by eigenvectors of ##\phi## The...
30. ### User-defined orthonormal basis

Does anybody know how to create a orthonormal basis, i.e. a matrix containing orthogonal vectors of norm 1, out of a given direction (normalised vector or versor) in a space with dimension N>3? With "out of a given direction", I mean that the resulting basis would have the first vector equal...
31. ### Finding an orthonormal basis of V

I've done most of this question apart from the very last bit. I have an answer to the very last bit, but it doesn't use any of my previously proved statements and I think they probably mean me to deduce from what I already have. Homework Statement Let V be the finite-dimensional vector...
32. ### Orthonormal basis vectors for polar coordinate system

Firstly; is there a difference between the "regular" polar coordinates that use \theta and r to describe a point (the one where the point (\sqrt{2}, \frac{\pi}{4}) equals (1, 1) in rectangular coordinates) and the ones that use the orthonormal basis vectors \hat{e}_r and...
33. ### Complex Number Orthonormal Basis.

Find an orthonormal basis for P2(ℂ) with respect to the inner product: <p(x),q(x)> = p(0)q(0) + p(i)q(i) + p(2i)q(2i) the q(x) functions are suppose to be the conjugates I just don't know how to write it on the computer Attempt: This is where I'm having trouble. So usually I'm given...
34. ### Orthonormal basis functions for L^2(R)

Hello, are there sets of functions that form an orthonormal basis for the space of square integrable functions over the reals L2(ℝ)? According to Wikipedia the hermite polynomials form an orthogonal basis (w.r.t. to a certain weight function) for L2(ℝ). So I guess it would be a matter of...
35. ### Circular coordinate space using an orthonormal basis

If we have any two orthonormal vectors A and B in R^2 and we wish to describe the circle they create under rigid rotation (i.e. they rotate at a fixed point and their length is preserved), how can we describe any point along this (unit) circle using a linear combination of A and B? I was...
36. ### Finding orthonormal basis for the intersection of the subspaces

Homework Statement Homework Equations can someone help me to solve this problem? The Attempt at a Solution I couldn't even approach
37. ### Inner product space and orthonormal basis.

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

Homework Statement The four functions v0 = 1; v1 = t; v2 = t^2; v3 = t^3 form a basis for the vector space of polynomials of degree 3. Apply the Gram-Schmidt procedure to find an orthonormal basis with respect to the inner product: < f ; g >= (1/2)\int 1-1 f(t)g(t) dtHomework Equations ui =...
39. ### Finding an orthonormal basis for a reproducing kernel Hilbert space.

Hello all, I'm currently working on a problem in which I'm attempting to characterize a centered Gaussian random process \xi(x) on a manifold M given a known covariance function C(x,x') for that process. My current approach is to find a series expansion \$\xi(x) = \sum_{n=1}^{\infty} X_n...
40. ### Orthonormal basis => vanishing Riemann curvature tensor

Hey! If a (pseudo) Riemannian manifold has an orthonormal basis, does it mean that Riemann curvature tensor vanishes? Orthonormal basis means that the metric tensor is of the form (g_{\alpha\beta}) = \text{diag}(-1,+1,+1,+1) what causes Christoffel symbols to vanish and puts Riemann...
41. ### Linear Algebra: Orthonormal Basis

Homework Statement Find an orthonormal basis for the subspace of R^4 that is spanned by the vectors: (1,0,1,0), (1,1,1,0), (1,-1,0,1), (3,4,4,-1) The Attempt at a Solution When I try to use the Gram-Schmidt process, I am getting (before normalization): (1,0,1,0), (0,1,0,0), (1,0,-1,2)...
42. ### Find the orthonormal basis

Homework Statement Consider R3 together with the standard inner product. Let A = 1 1 −1 2 1 3 1 2 −6 (a) Use the Gram-Schmidt process to find an orthonormal basis S1 for null(A), and an orthonormal basis S2 for col(A). (b) Note that S = S1 ∪ S2 is a basis for R3. Use the the...
43. ### Orthonormal Basis Homework: Gram-Schmidt Process w/ Inner Product

Homework Statement Hi, i am applying the gram-schmidt procedure to a basis of {1,2x,3x^2} with inner product <p,q> = \int p(x)q(x) from 0 to 1. i am unsure what to do with the inner product Homework Equations The Attempt at a Solution I have followed the procedure i have for...
44. ### Find Orthonormal Basis of R3: u1,u2,u3

Homework Statement Note: the vectors are column vectors, not row vectors. Latex is not working for me right now. Find an orthonormal basis u1, u2, u3 of R3 such that span(u1) = span [1 2 3] and span(u1,u2) = span { [1 2 3], [1 1 -1] } Homework Equations The Attempt at...
45. ### Matrix Elements of Operators & Orthonormal Basis Sets

So, the rule for finding the matrix elements of an operator is: \langle b_i|O|b_j\rangle Where the "b's" are vector of the basis set. Does this rule work if the basis is not orthonormal? Because I was checking this with regular linear algebra (in R3) (finding matrix elements of linear...
46. ### How do I find the orthonormal basis for the intersection of subspaces U and V?

Homework Statement Hi, i am trying to do the question on the image, Can some one help me out with the steps. [PLAIN]http://img121.imageshack.us/img121/6818/algebra0.jpg [Broken] Solution in the image is right but my answer is so off from the current one. Homework Equations...
47. ### Physical meaning of orthonormal basis

I want to know what orthonormal basis or transformation physically means. Can anyone please explain me with a practical example? I prefer examples as to where it is put to use practically rather than examples with just numbers..
48. ### Finding Orthonormal Basis of Hilbert Space wrt Lattice of Subspaces

I have a Hilbert space H; given a closed subspace U of H let PU denote the orthogonal projection onto U. I also have a lattice L of closed subspaces of H, such that for all U and U' in L, PU and PU' commute. The problem is to find an orthonormal basis B of H, such that for every element b of B...
49. ### Inner product as integral, orthonormal basis

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...
50. ### Orthonormal basis spanned by 2 matrices

Homework Statement Let M1 = [1 1] and M2 = [-3 -2] ________[1 -1]_________[ 1 2] Consider the inner product <A,B> = trace(transpose(A)B) in the vector space R2x2 of 2x2 matrices. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of R2x2 spanned by the...