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.
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...
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...
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...
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...
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 \\...
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...
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...
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.
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...
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|...
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!
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...
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...
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...
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...
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...
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...
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...
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##...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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 =...
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...
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...
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)...
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...
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...
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...
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...
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...
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..
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...
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...
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...