Linear algebra

Orthogonal transformation

In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product. That is, for each pair u, v of elements of V, we have Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles between them. In particular, orthogonal transformations map orthonormal bases to orthonormal bases. Orthogonal transformations are injective: if then , hence , so the kernel of is trivial. Orthogonal transformations in two- or three-dimensional Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations). Reflections are transformations that reverse the direction front to back, orthogonal to the mirror plane, like (real-world) mirrors do. The matrices corresponding to proper rotations (without reflection) have a determinant of +1. Transformations with reflection are represented by matrices with a determinant of −1. This allows the concept of rotation and reflection to be generalized to higher dimensions. In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix. Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of V. The columns of the matrix form another orthonormal basis of V. If an orthogonal transformation is invertible (which is always the case when V is finite-dimensional) then its inverse is another orthogonal transformation. Its matrix representation is the transpose of the matrix representation of the original transformation. (Wikipedia).

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11H Orthogonal Projection of a Vector

The orthogonal projection of one vector along another.

From playlist Linear Algebra

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11J Orthogonal Projection of a Vector

The orthogonal projection of one vector along another.

From playlist Linear Algebra

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Orthogonal Projections

In this video, I define the concept of orthogonal projection of a vector on a line (and on more general subspaces), derive a very nice formula for it, and show why orthogonal projections are so useful. You might even see the hugging formula again. Enjoy! This is the second part of the ort

From playlist Orthogonality

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11I Orthogonal Projection of a Vector

The Orthogonal Projection of one vector along another.

From playlist Linear Algebra

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Linear Algebra 21j: Two Geometric Interpretations of Orthogonal Matrices

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Part 3 Linear Algebra: Linear Transformations

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Linear Algebra 23a: Polar Decomposition - A Product of an Orthogonal and Symmetric Matrices

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Part 3 Linear Algebra: Linear Transformations

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Orthogonal Transformations 1: 2x2 Case

Linear Algebra: Let A be a 2x2 orthogonal matrix. A general form for A is given, and we show that A corresponds to either a rotation or reflection of the plane. (Added: Minor edit to reflections.)

From playlist MathDoctorBob: Linear Algebra I: From Linear Equations to Eigenspaces | CosmoLearning.org Mathematics

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Inverse Transformation

In this video, I calculate the inverse of a linear transformation, by first writing it as a matrix A and then calculating A^-1 This illustrates the nice relationship between linear transformations and matrices. Enjoy! Check out my Linear Transformations Playlist: https://www.youtube.com/p

From playlist Linear Transformations

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Linear Algebra 22a: Introduction to Orthoscaling (aka Symmetric) Transformations

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Part 3 Linear Algebra: Linear Transformations

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22: Rigid body rotation - Part 2

Jacob Linder: 15.02.12, Classical Mechanics (TFY4345), v2012 NTNU A full textbook covering the material in the lectures in detail can be downloaded for free here: http://bookboon.com/en/introduction-to-lagrangian-hamiltonian-mechanics-ebook

From playlist NTNU: TFY 4345 - Classical Mechanics | CosmoLearning Physics

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Francesco Mezzadri: Moments of Random Matrices and Hypergeometric Orthogonal Polynomials

We establish a new connection between moments of n×n random matrices $X_{n}$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s\in\mathbb{C}$, whose analytic structure we describe completely

From playlist Jean-Morlet Chair - Grava/Bufetov

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AGACSE2021 Joan Lasenby - GA approach to orthogonal transformations in signal and image processing.

Professor Joan Lasenby from Cambridge University on a Geometric Algebra approach to orthogonal transformations and their use in signal and image processing.

From playlist AGACSE2021

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Stéphane Mallat: A Wavelet Zoom to Analyze a Multiscale World

Abstract: Complex physical phenomena, signals and images involve structures of very different scales. A wavelet transform operates as a zoom, which simplifies the analysis by separating local variations at different scales. Yves Meyer found wavelet orthonormal bases having better propertie

From playlist Abel Lectures

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21: Rigid body rotation

Jacob Linder: 15.02.12, Classical Mechanics (TFY4345), v2012 NTNU A full textbook covering the material in the lectures in detail can be downloaded for free here: http://bookboon.com/en/introduction-to-lagrangian-hamiltonian-mechanics-ebook

From playlist NTNU: TFY 4345 - Classical Mechanics | CosmoLearning Physics

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Wavelets and Multiresolution Analysis

This video discusses the wavelet transform. The wavelet transform generalizes the Fourier transform and is better suited to multiscale data. Book Website: http://databookuw.com Book PDF: http://databookuw.com/databook.pdf These lectures follow Chapter 2 from: "Data-Driven Science an

From playlist Data-Driven Science and Engineering

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Another example of a projection matrix | 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/orthogonal-projections/v/lin-alg-another-example-of-a-projection-matrix Figuring out the transformation matrix for a projection onto

From playlist Alternate coordinate systems (bases) | Linear Algebra | Khan Academy

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Mod-01 Lec-01 Introduction and Overview

Advanced Numerical Analysis by Prof. Sachin C. Patwardhan,Department of Chemical Engineering,IIT Bombay.For more details on NPTEL visit http://nptel.ac.in

From playlist IIT Bombay: Advanced Numerical Analysis | CosmoLearning.org

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Linear Algebra 7.1 Orthogonal Matrices

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.

From playlist Linear Algebra

Related pages

Orthogonal matrix | Dimension (vector space) | Unitary transformation | Improper rotation | Linear algebra | Determinant | Inner product space | Real number | Orthonormal basis | Kernel (linear algebra) | Matrix (mathematics) | Euclidean space | Reflection (mathematics) | Rotation (mathematics)