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.
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...
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 +...
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...
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...
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...
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...
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...
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...
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...
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...
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)...
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...
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
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...
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...
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.