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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
Orthogonal + orthonormal vectors
Orthogonal + orthonormal vectors
Published: 2014/03/19
Channel: Dr Chris Tisdell
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
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
Orthogonal Set of Functions ( Fourier Series )
Orthogonal Set of Functions ( Fourier Series )
Published: 2012/10/31
Channel: patrickJMT
Orthogonality of wavefunctions
Orthogonality of wavefunctions
Published: 2013/10/27
Channel: BU Chem
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
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
ChaosBook.org chapter Continuous symmetry factorization - Character orthonormality
ChaosBook.org chapter Continuous symmetry factorization - Character orthonormality
Published: 2015/04/20
Channel: birdtracks
Orthogonality and Orthonormality - Linear Algebra
Orthogonality and Orthonormality - Linear Algebra
Published: 2017/03/25
Channel: Engineer Thileban Explains
Quantum Chemistry 4.9 - Orthogonality
Quantum Chemistry 4.9 - Orthogonality
Published: 2016/08/14
Channel: TMP Chem
What does orthonormality mean?
What does orthonormality mean?
Published: 2018/01/01
Channel: What Does That Mean?
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
Definition of Orthonormal
Definition of Orthonormal
Published: 2017/02/14
Channel: James Johns
Fourier Series: Example of Orthonormal Set of Functions
Fourier Series: Example of Orthonormal Set of Functions
Published: 2011/03/02
Channel: MathDoctorBob
Gram Schmidt Method, Orthogonal and Orhonormal Basis Example
Gram Schmidt Method, Orthogonal and Orhonormal Basis Example
Published: 2015/12/08
Channel: MrClean1796
Orthonormal function Meaning
Orthonormal function Meaning
Published: 2015/04/27
Channel: ADictionary
Lec 15 - Orthonormality
Lec 15 - Orthonormality
Published: 2018/01/28
Channel: MATHEMATICAL METHODS AND TECHNIQUES IN SIGNAL PROCESSING
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
How to say "orthonormality"! (High Quality Voices)
How to say "orthonormality"! (High Quality Voices)
Published: 2017/02/08
Channel: WordBox
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
Orthonormality of Standing Waves
Orthonormality of Standing Waves
Published: 2010/04/22
Channel: wolframmathematica
Orthonormality - Linear Algebra - G3
Orthonormality - Linear Algebra - G3
Published: 2017/08/17
Channel: Tom Roby
Tutorial Q73 -- Orthonormal system, Example
Tutorial Q73 -- Orthonormal system, Example
Published: 2014/09/05
Channel: Denis Potapov
j   Linear Algebra Tutor   Vol 1   Sect 10   Orthogonal And Orthonormal Vectors
j Linear Algebra Tutor Vol 1 Sect 10 Orthogonal And Orthonormal Vectors
Published: 2017/07/17
Channel: Video Storage
How to Pronounce orthonormality - American English
How to Pronounce orthonormality - American English
Published: 2018/02/05
Channel: HowToPronounce16
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
How to Pronounce Orthonormality
How to Pronounce Orthonormality
Published: 2013/12/24
Channel: Emma Saying
How To Pronounce Orthonormality
How To Pronounce Orthonormality
Published: 2017/10/15
Channel: Emma Saying
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
Orthonormal Substitution
Orthonormal Substitution
Published: 2017/11/24
Channel: Jesper Kampmann Larsen
Math 060 Fall 2017 111317 Orthonormal Bases
Math 060 Fall 2017 111317 Orthonormal Bases
Published: 2017/12/18
Channel: Winston Ou
4.3 Mecánica Cuántica - Orthogonality and orthonormal basis
4.3 Mecánica Cuántica - Orthogonality and orthonormal basis
Published: 2018/02/17
Channel: Discovery Class
daglin 1071 orthogonal basic of R2 easily converted to orthonormal  finding orthogonal basis given n
daglin 1071 orthogonal basic of R2 easily converted to orthonormal finding orthogonal basis given n
Published: 2017/12/11
Channel: dave smith
Linear Algebra: Lecture 30: eigenspace decompositions, orthonormality
Linear Algebra: Lecture 30: eigenspace decompositions, orthonormality
Published: 2015/04/06
Channel: James Cook
Orthogonal Matrices - Rank of Matrix - Engineering Mathematics 1 - First Year Engineering
Orthogonal Matrices - Rank of Matrix - Engineering Mathematics 1 - First Year Engineering
Published: 2015/10/12
Channel: Ekeeda
Vectors : - ( Orthonormal Vectors ) - 26.
Vectors : - ( Orthonormal Vectors ) - 26.
Published: 2018/02/24
Channel: Ahamed Ameen
Orthonormalization
Orthonormalization
Published: 2015/01/10
Channel: John Harkless
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
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
How to say "orthonormal"! (High Quality Voices)
How to say "orthonormal"! (High Quality Voices)
Published: 2017/02/09
Channel: WordBox
11.3.2 Orthonormal Vectors Part 5
11.3.2 Orthonormal Vectors Part 5
Published: 2014/04/20
Channel: LAFF Linear Algebra - Foundations to Frontiers (www.ulaff.net)
How to Pronounce Orthonormality
How to Pronounce Orthonormality
Published: 2015/05/30
Channel: Pronunciation Guide
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
Completeness and Orthogonality
Completeness and Orthogonality
Published: 2014/09/12
Channel: Dr. Underwood's Physics YouTube Page
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
daglin 7873 orthonormal basis of R 2  orthonormalizon a given basis of R 2
daglin 7873 orthonormal basis of R 2 orthonormalizon a given basis of R 2
Published: 2016/12/01
Channel: dave smith
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WIKIPEDIA ARTICLE

From Wikipedia, the free encyclopedia
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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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