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Introduction to orthonormal bases | Linear Algebra | Khan Academy
Introduction to orthonormal bases | Linear Algebra | Khan Academy
Published: 2009/11/12
Channel: Khan Academy
Linear Algebra: Orthonormal Basis
Linear Algebra: Orthonormal Basis
Published: 2014/06/27
Channel: Worldwide Center of Mathematics
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Orthogonal + orthonormal vectors
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Channel: Dr Chris Tisdell
11.3.2 Orthonormal Vectors Part 1
11.3.2 Orthonormal Vectors Part 1
Published: 2014/04/20
Channel: LAFF Linear Algebra - Foundations to Frontiers (www.ulaff.net)
L21.2 Orthonormality of spherical harmonics.
L21.2 Orthonormality of spherical harmonics.
Published: 2017/07/05
Channel: MIT OpenCourseWare
How to normalize combinations of orthonormal wavefunctions - Real chemistry
How to normalize combinations of orthonormal wavefunctions - Real chemistry
Published: 2017/01/27
Channel: Real Chemistry
Lecture 6 - Change of Orthonormal Basis
Lecture 6 - Change of Orthonormal Basis
Published: 2017/10/05
Channel: Calvin Stewart
10  Orthogonal and Orthonormal Vectors
10 Orthogonal and Orthonormal Vectors
Published: 2016/02/03
Channel: 1234ep
Gram Schmidt Method, Orthogonal and Orhonormal Basis Example
Gram Schmidt Method, Orthogonal and Orhonormal Basis Example
Published: 2015/12/08
Channel: MrClean1796
Definition of Orthonormal
Definition of Orthonormal
Published: 2017/02/14
Channel: James Johns
Applied Linear: orthonormality in Euclidean space, 3-7-17, part 1
Applied Linear: orthonormality in Euclidean space, 3-7-17, part 1
Published: 2017/03/08
Channel: James Cook
Linear Algebra 10: Construction of a Orthonormal Basis (Ch4 Pr3)
Linear Algebra 10: Construction of a Orthonormal Basis (Ch4 Pr3)
Published: 2016/02/23
Channel: MathsStatsUNSW
Orthonormal Basis, and Gram Schidmt Process
Orthonormal Basis, and Gram Schidmt Process
Published: 2017/05/17
Channel: WeSolveThem
The Gram-Schmidt Process
The Gram-Schmidt Process
Published: 2013/11/01
Channel: Lorenzo Sadun
Orthogonality and Orthonormality | The Complete Linear Algebra Study Guide
Orthogonality and Orthonormality | The Complete Linear Algebra Study Guide
Published: 2016/12/02
Channel: Milosz Kowal
Orthonormality - Linear Algebra - G3
Orthonormality - Linear Algebra - G3
Published: 2017/08/17
Channel: Tom Roby
orthonormal orthogonal matrices
orthonormal orthogonal matrices
Published: 2014/09/26
Channel: Gabe Bentz
Quantum Chemistry 6.11 - Orthonormality of Spherical Harmonics
Quantum Chemistry 6.11 - Orthonormality of Spherical Harmonics
Published: 2016/08/14
Channel: TMP Chem
Linear Algebra: Lecture 30: eigenspace decompositions, orthonormality
Linear Algebra: Lecture 30: eigenspace decompositions, orthonormality
Published: 2015/04/06
Channel: James Cook
(7/40) EIGENFUNCTIONS,EIGENVALUES,NORMALIZED,ORTHOGONAL AND ORTHONORMAL EIGENFUNCTIONS
(7/40) EIGENFUNCTIONS,EIGENVALUES,NORMALIZED,ORTHOGONAL AND ORTHONORMAL EIGENFUNCTIONS
Published: 2016/01/10
Channel: Chitra Thomas
Orthonormal Bases
Orthonormal Bases
Published: 2017/05/07
Channel: Sheldon Axler
Orthogonal Set of Functions ( Fourier Series )
Orthogonal Set of Functions ( Fourier Series )
Published: 2012/10/31
Channel: patrickJMT
Topics In Quantum Mechanics Video #7: Example Of Orthonormal Function
Topics In Quantum Mechanics Video #7: Example Of Orthonormal Function
Published: 2011/05/01
Channel: TheDigitalUniversity
Orthogonality and Orthonormality - Linear Algebra
Orthogonality and Orthonormality - Linear Algebra
Published: 2017/03/25
Channel: Engineer Thileban Explains
ChaosBook.org chapter Continuous symmetry factorization - The Great Orthonormality Theorem
ChaosBook.org chapter Continuous symmetry factorization - The Great Orthonormality Theorem
Published: 2015/04/19
Channel: birdtracks
Finding norm and the normalized function and discuss orthogonality and orthonormal functions 12-8-2
Finding norm and the normalized function and discuss orthogonality and orthonormal functions 12-8-2
Published: 2016/03/13
Channel: Marx Academy
Applied Linear: orthonormality in Euclidean space, 3-7-17, part 2
Applied Linear: orthonormality in Euclidean space, 3-7-17, part 2
Published: 2017/03/08
Channel: James Cook
Projections onto subspaces with orthonormal bases | Linear Algebra | Khan Academy
Projections onto subspaces with orthonormal bases | Linear Algebra | Khan Academy
Published: 2009/11/12
Channel: Khan Academy
Lesson 10 - Orthogonal And Orthonormal Vectors (Linear Algebra)
Lesson 10 - Orthogonal And Orthonormal Vectors (Linear Algebra)
Published: 2016/08/19
Channel: mathtutordvd
Linear Algebra- Determining Whether a Set of Vectors is Orthogonal, Orthonormal, or Neither
Linear Algebra- Determining Whether a Set of Vectors is Orthogonal, Orthonormal, or Neither
Published: 2016/07/14
Channel: Kelley's Math & Stats Help
Linear Algebra: April 4, orthonormality and Gram Schmidt Algorithm
Linear Algebra: April 4, orthonormality and Gram Schmidt Algorithm
Published: 2016/04/18
Channel: James Cook
Quantum Chemistry 6.11 - Orthonormality of Spherical Harmonics (Old Version)
Quantum Chemistry 6.11 - Orthonormality of Spherical Harmonics (Old Version)
Published: 2014/02/09
Channel: TMP Chem
Fourier Series: Example of Orthonormal Set of Functions
Fourier Series: Example of Orthonormal Set of Functions
Published: 2011/03/02
Channel: MathDoctorBob
Orthogonality of wavefunctions
Orthogonality of wavefunctions
Published: 2013/10/27
Channel: BU Chem
Orthonormal function Meaning
Orthonormal function Meaning
Published: 2015/04/27
Channel: ADictionary
Orthonormality of Standing Waves
Orthonormality of Standing Waves
Published: 2010/04/22
Channel: wolframmathematica
Mod-01 Lec-29 Total Orthonormal Sets And Sequences
Mod-01 Lec-29 Total Orthonormal Sets And Sequences
Published: 2012/06/19
Channel: nptelhrd
Find Eigenvalues, Orthonormal eigenvectors , Diagonazible - Linear Algebra
Find Eigenvalues, Orthonormal eigenvectors , Diagonazible - Linear Algebra
Published: 2017/04/07
Channel: Engineer Thileban Explains
grade9 orthonormal system exercise
grade9 orthonormal system exercise
Published: 2017/05/17
Channel: Ibrahim Dhaini
Coordinates with respect to orthonormal bases | Linear Algebra | Khan Academy
Coordinates with respect to orthonormal bases | Linear Algebra | Khan Academy
Published: 2009/11/12
Channel: Khan Academy
Mod-01 Lec-23 Projection Theorem, Orthonormal Sets and Sequences
Mod-01 Lec-23 Projection Theorem, Orthonormal Sets and Sequences
Published: 2012/06/19
Channel: nptelhrd
11.3.2 Orthonormal Vectors Part 8
11.3.2 Orthonormal Vectors Part 8
Published: 2014/04/20
Channel: LAFF Linear Algebra - Foundations to Frontiers (www.ulaff.net)
How to say "orthonormality"! (High Quality Voices)
How to say "orthonormality"! (High Quality Voices)
Published: 2017/02/08
Channel: WordBox
Orthonormal basis vectors
Orthonormal basis vectors
Published: 2017/01/03
Channel: Robert Davie
Week 9 - Orthonormal bases
Week 9 - Orthonormal bases
Published: 2014/06/16
Channel: mathapptician
Mathematics of Quantum 11 (Orthonormal Basis, kronecker delta function)
Mathematics of Quantum 11 (Orthonormal Basis, kronecker delta function)
Published: 2015/12/03
Channel: Kaveh Mozafari
One Advantage of Using an Orthonormal Basis
One Advantage of Using an Orthonormal Basis
Published: 2011/11/12
Channel: refrigeratormathprof
Quantum Chemistry 4.9 - Orthogonality
Quantum Chemistry 4.9 - Orthogonality
Published: 2016/08/14
Channel: TMP Chem
Representation Theory 6, Standard Inner Product, Orthogonal and Orthonormal
Representation Theory 6, Standard Inner Product, Orthogonal and Orthonormal
Published: 2013/09/08
Channel: LadislauFernandes
LA- 50 - Orthonormal Basis And Gram-Shmidt Process
LA- 50 - Orthonormal Basis And Gram-Shmidt Process
Published: 2017/10/15
Channel: Ahmed Fathi
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WIKIPEDIA ARTICLE

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In linear algebra, two vectors in an inner product space are orthonormal, if they are orthogonal and unit vectors. A set of vectors form an orthonormal set, if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.

Intuitive overview[edit]

The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be perpendicular if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero.

Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces. In Cartesian space, the norm of a vector is the square root of the vector dotted with itself. That is,

Many important results in linear algebra deal with collections of two or more orthogonal vectors. But often, it is easier to deal with vectors of unit length. That is, it often simplifies things to only consider vectors whose norm equals 1. The notion of restricting orthogonal pairs of vectors to only those of unit length is important enough to be given a special name. Two vectors which are orthogonal and of length 1 are said to be orthonormal.

Simple example[edit]

What does a pair of orthonormal vectors in 2-D Euclidean space look like?

Let u = (x1, y1) and v = (x2, y2). Consider the restrictions on x1, x2, y1, y2 required to make u and v form an orthonormal pair.

  • From the orthogonality restriction, uv = 0.
  • From the unit length restriction on u, ||u|| = 1.
  • From the unit length restriction on v, ||v|| = 1.

Expanding these terms gives 3 equations:

Converting from Cartesian to polar coordinates, and considering Equation and Equation immediately gives the result r1 = r2 = 1. In other words, requiring the vectors be of unit length restricts the vectors to lie on the unit circle.

After substitution, Equation becomes . Rearranging gives . Using a trigonometric identity to convert the cotangent term gives

It is clear that in the plane, orthonormal vectors are simply radii of the unit circle whose difference in angles equals 90°.

Definition[edit]

Let be an inner-product space. A set of vectors

is called orthonormal if and only if

where is the Kronecker delta and is the inner product defined over .

Significance[edit]

Orthonormal sets are not especially significant on their own. However, they display certain features that make them fundamental in exploring the notion of diagonalizability of certain operators on vector spaces.

Properties[edit]

Orthonormal sets have certain very appealing properties, which make them particularly easy to work with.

  • Theorem. If {e1, e2,...,en} is an orthonormal list of vectors, then

Existence[edit]

  • Gram-Schmidt theorem. If {v1, v2,...,vn} is a linearly independent list of vectors in an inner-product space , then there exists an orthonormal list {e1, e2,...,en} of vectors in such that span(e1, e2,...,en) = span(v1, v2,...,vn).

Proof of the Gram-Schmidt theorem is constructive, and discussed at length elsewhere. The Gram-Schmidt theorem, together with the axiom of choice, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors. What results is a deep relationship between the diagonalizability of an operator and how it acts on the orthonormal basis vectors. This relationship is characterized by the Spectral Theorem.

Examples[edit]

Standard basis[edit]

The standard basis for the coordinate space Fn is

{e1, e2,...,en}   where    e1 = (1, 0, ..., 0)
   e2 = (0, 1, ..., 0)
   en = (0, 0, ..., 1)

Any two vectors ei, ej where i≠j are orthogonal, and all vectors are clearly of unit length. So {e1, e2,...,en} forms an orthonormal basis.

Real-valued functions[edit]

When referring to real-valued functions, usually the inner product is assumed unless otherwise stated. Two functions and are orthonormal over the interval if

Fourier series[edit]

The Fourier series is a method of expressing a periodic function in terms of sinusoidal basis functions. Taking C[−π,π] to be the space of all real-valued functions continuous on the interval [−π,π] and taking the inner product to be

It can be shown that

forms an orthonormal set.

However, this is of little consequence, because C[−π,π] is infinite-dimensional, and a finite set of vectors cannot span it. But, removing the restriction that n be finite makes the set dense in C[−π,π] and therefore an orthonormal basis of C[−π,π].

See also[edit]

References[edit]

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