Functional analysis | Fourier analysis | Linear algebra
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for 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 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 arises in this fashion. For a general inner product space an orthonormal basis can be used to define normalized orthogonal coordinates on 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 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 an orthonormal basis for is an orthonormal set of vectors with the property that every vector in 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 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 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 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 A different generalisation is to pseudo-inner product spaces, finite-dimensional vector spaces equipped with a non-degenerate symmetric bilinear form known as the metric tensor. In such a basis, the metric takes the form with positive ones and negative ones. (Wikipedia).
Math 060 Fall 2017 111317C Orthonormal Bases
Motivation: how to obtain the coordinate vector with respect to a given basis? Definition: orthogonal set. Example. Orthogonal implies linearly independent. Orthonormal sets. Example of an orthonormal set. Definition: orthonormal basis. Properties of orthonormal bases. Example: Fou
From playlist Course 4: Linear Algebra (Fall 2017)
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Learn the basics of Linear Algebra with this series from the Worldwide Center of Mathematics. Find more math tutoring and lecture videos on our channel or at http://centerofmath.org/ More on unit vectors: https://www.youtube.com/watch?v=C6EYJVBYXIo
From playlist Basics: Linear Algebra
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From playlist Linear Algebra Done Right
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From playlist MATH2501 - Linear Algebra
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From playlist Orthogonal and Orthonormal Sets of Vectors
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Math 060 Fall 2017 111517C Orthonormal Bases, Orthogonal Matrices, and Method of Least Squares
Definition of orthogonal matrices. Example: rotation matrix. Properties: Q orthogonal if and only if its transpose is its inverse. Q orthogonal implies it is an isometry; that it is isogonal (preserves angles). Theorem: How to find, given a vector in an inner product space, the closest
From playlist Course 4: Linear Algebra (Fall 2017)
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MIT 18.102 Introduction to Functional Analysis, Spring 2021 Instructor: Dr. Casey Rodriguez View the complete course: https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/ YouTube Playlist: https://www.youtube.com/watch?v=Yb69dAq4uh8&list=PLUl4u3cNGP63micsJp_
From playlist MIT 18.102 Introduction to Functional Analysis, Spring 2021
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My notes are available at http://asherbroberts.com/ (so you can write along with me). Elementary Linear Algebra: Applications Version 12th Edition by Howard Anton, Chris Rorres, and Anton Kaul A. Roberts is supported in part by the grants NSF CAREER 1653602 and NSF DMS 2153803.
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Coordinates with respect to orthonormal bases | Linear Algebra | Khan Academy
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/linear-algebra/alternate-bases/orthonormal-basis/v/linear-algebra-coordinates-with-respect-to-orthonormal-bases Seeing that orthonormal bases make for good coordin
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From playlist MIT 18.102 Introduction to Functional Analysis, Spring 2021
The Gram-Schmidt process | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/linear-algebra/alternate-bases/orthonormal-basis/v/linear-algebra-the-gram-schmidt-process Finding an orthonormal basis for a subspace using the Gram-Schmidt Proce
From playlist Alternate coordinate systems (bases) | Linear Algebra | Khan Academy
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