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. Z

    Linear transformation of an orthonormal basis

    Homework Statement Consider a linear transformation L from Rm to Rn. Show that there is an orthonormal basis {v1,...,vm} of Rm to Rn such that the vectors {L(v1),...,L(vm)} are orthogonal. Note that some of the vectors L(vi) may be zero. HINT: Consider an orthonormal basis {v1,...,vm} for...
  2. G

    Orthonormal basis for subsets of C^3

    We want to find a basis for W and W_perpendicular for W=span({(i,0,1)}) =Span({w1}) in C^3 a vector x =(a,b,c) in W_perp satisfies <w1,x> = 0 => ai + c = 0 => c=-ai Thus a vector x in W_perp is x = (a,b,-ai) So an orthonormal basis in W would be simply w1/norm(w1) ...but the norm(w1)=0 (i^2 +...
  3. A

    Proving Orthonormal Basis for an Orthogonal Matrix

    Homework Statement Prove: if an n × n matrix A is orthogonal (column vectors are orthonormal), then the columns form an orthonormal basis for R^n. (with respect to the standard Euclidean inner product [= the dot product]). Homework Equations None. The Attempt at a Solution I...
  4. M

    Calculating Eigenkets from Matrix w/ Orthonormal Basis

    How am i supposed to write eigenkets of an operator whose matrix is given to me given that the two ket vectors form an orthonormal basis .
  5. P

    Orthonormal basis in fermi coordinates

    please help in this problem what are these basis and what are there there properties.how i can i put there values to solve my problems. ˆB = ( 1 2  + +)er er + ( 1 2  − +)e e + (× + !)er e + (× − !)e er . (4.13) where eµ are co-frame basis satisfying...
  6. D

    Orthonormal Basis Homework: True/False

    Homework Statement True/False: The set of vectors B={(-1,-1,1,1),(1,0,0,0),(0,1,0,0),(-1,-1,1,-1)} is an orthonormal basis for Euclidean 4-space \mathbb{R}^4. Homework Equations NoneThe Attempt at a Solution I said false because \langle (-1,-1,1,1),(-1,-1,1,1) \rangle =2\ne1, which shows...
  7. A

    Orthonormal Basis: Showing Wave Functions are Orthonormal

    Homework Statement Hey guys. http://img39.imageshack.us/img39/2345/27760913.jpg I need to show that these wave functions are orthonormal. I'm a bit confuse, what's i and what's j? I mean, do I need to take both of the functions, put them in the integral and to show that the result...
  8. S

    Gram-Schmidt orthonormal basis

    Homework Statement Let R^3 have the inner product <u, v> = u1v1 + 2u2v2 + 3u3v3. Use the Gram-Schmidt process to convert u1=(1,1,1), u2 = (1,1,0), u3 = (1,0,0) into a normal orthonormal basis Homework Equations I know the process for the orthonomoral converasion. I have no problem...
  9. T

    Complete to orthonormal basis question

    i got these vectors which are othronormal v1 (1/2,-1/2,1/2,-1/2) v2 (-1/2,1/2,1/2,-1/2) i need to compete them into orthonormal basis i did row reduction on them and added these independant vectors to the group v3(1,0,0,0) v4(0,1,0,0) now all four vectors are independant...
  10. S

    How can the orthonormal basis of four vectors be found?

    Homework Statement How can I find the orthonormal basis of four vectors? The vectors are: (0, 3, 0, 4), (4, 0, 3, 0), (4, 3, 3, 4) and (4, -3, 3, -4). The Attempt at a Solution I am not sure, whether I should use Gram-Schmidt process or the process of finding eigenvalues, eigenvectors and...
  11. D

    How Do You Find an Orthonormal Basis for a Subspace With a Sum Condition?

    Homework Statement Find an orthonormal basis for the subspace of R^3 consisting of all vectors (a, b, c) such that a + b + c = 0. Homework Equations The Attempt at a Solution I know how to find an orthonormal basis just for R^3 by taking the standard basis vectors (1, 0, 0), (0...
  12. A

    Solving Orthonormal Basis of Eigenvectors for Matrix A

    Homework Statement My problem is I am getting a different answer than what MATLAB is giving me and I cannot determine why. Plz advise. Find an orthonormal basis of eigenvectors for matrix A= [3 2; 2 1] (using MATLAB notation- I couldn't figure out how to put in proper matrix notation)...
  13. F

    Orthonormal basis and operators

    I hope this is the forum to ask this question. We all know that the eigenvectors of a Hermitian operator form an orthonormal basis. But is the opposite true as well. Are the vectors of an orthonormal basis always the eigenvectors of some Hermitian operator? Or do we need added restrictions to...
  14. P

    Is A Required to Have Orthonormal Basis?

    If A*A=AA* than A is a normal matrix. But does A must also have an orthonormal basis?
  15. G

    What is the Orthonormal Basis of the Plane x - 4y - z = 0?

    I need to find the Orthonormal Basis of this plane: x - 4y -z = 0 I know the result will be the span of two vectors but I'm not sure where to start. Any hints? Thanks, Gab
  16. B

    Exploring Orthonormal Basis for 2x2 Complex Matrices

    My other problem is: Consider now the space of 2x2 complex matrices. Show that the Pauli Matrices |I>= 1 0 0 1 |sigma x>= 0 1 1 0 |sigma y>= 0 -i i 0 |sigma z>= 1 0 0 -1 form an orthonormal basis for this space...
  17. B

    Orthogonal projection, orthonormal basis, coordinate vector of the polynomial?

    Hey there I'm working on questions for a sample review for finals I'm stuck on these three I think I'm starting to confuse all the different theorem, I'm so lost please help 1) Find the coordinate vector of the polynomial p(x)=1+x+x^2 relative to the following basis of P2: p1=1+x...
  18. G

    Help with making an orthonormal basis

    I'm wanting to form an orthonormal basis from two non-parallel vectors. a = \left(\begin{array}{cc}3 & 4\end{array}\right) b = \left(\begin{array}{cc}2 & -6\end{array}\right) Could someone please walk me through the calculations needed? Much appreciated.
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