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. nomadreid

    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. George Keeling

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  3. S

    I Orthonormal basis expression for ordinary contraction of a tensor

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  4. H

    I Proof that if T is Hermitian, eigenvectors form an orthonormal basis

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  5. M

    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. M

    MHB Orthonormal basis - Set of all isometries

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  7. The black vegetable

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    Is this correct? If not any hints on how to find Many thanks
  8. Jd_duarte

    I Orthonormal Basis of Wavefunctions in Hilbert Space

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  9. S

    Gram-Schmidt for 1, x, x^2 Must find orthonormal basis

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  10. Pushoam

    I Expanding a given vector into another orthonormal basis

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  11. D

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  12. L

    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. dreens

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  14. L

    Orthonormal basis of 1 forms for the rotating c metric

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  15. RJLiberator

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  16. Tony Stark

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  17. Tony Stark

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  18. A

    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. R

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  20. S

    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. C

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  22. D

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  23. C

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  24. Petrus

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  25. P

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  26. Sudharaka

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    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. A

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  28. M

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  29. F

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  30. M

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  31. U

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  32. V

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  33. T

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  34. mnb96

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  35. B

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  36. V

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  37. L

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  38. M

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    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. A

    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. P

    Orthonormal basis => vanishing Riemann curvature tensor

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  41. T

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  42. L

    Find the orthonormal basis

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  43. M

    Orthonormal Basis Homework: Gram-Schmidt Process w/ Inner Product

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  44. M

    Find Orthonormal Basis of R3: u1,u2,u3

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  45. A

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  46. I

    How do I find the orthonormal basis for the intersection of subspaces U and V?

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  47. L

    Physical meaning of orthonormal basis

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  48. A

    Finding Orthonormal Basis of Hilbert Space wrt Lattice of Subspaces

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  49. H

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  50. H

    Orthonormal basis spanned by 2 matrices

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