Transformation (function) | Matrices
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix rotates points in the xy plane counterclockwise through an angle θ with respect to the positive x axis about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R: If x and y are the endpoint coordinates of a vector, where x is cosine and y is sine, then the above equations become the trigonometric summation angle formulae. Indeed, a rotation matrix can be seen as the trigonometric summation angle formulae in matrix form. One way to understand this is say we have a vector at an angle 30° from the x axis, and we wish to rotate that angle by a further 45°. We simply need to compute the vector endpoint coordinates at 75°. The examples in this article apply to active rotations of vectors counterclockwise in a right-handed coordinate system (y counterclockwise from x) by pre-multiplication (R on the left). If any one of these is changed (such as rotating axes instead of vectors, a passive transformation), then the inverse of the example matrix should be used, which coincides with its transpose. Since matrix multiplication has no effect on the zero vector (the coordinates of the origin), rotation matrices describe rotations about the origin. Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In some literature, the term rotation is generalized to include improper rotations, characterized by orthogonal matrices with a determinant of −1 (instead of +1). These combine proper rotations with reflections (which invert orientation). In other cases, where reflections are not being considered, the label proper may be dropped. The latter convention is followed in this article. Rotation matrices are square matrices, with real entries. More specifically, they can be characterized as orthogonal matrices with determinant 1; that is, a square matrix R is a rotation matrix if and only if RT = R−1 and det R = 1. The set of all orthogonal matrices of size n with determinant +1 is a representation of a group known as the special orthogonal group SO(n), one example of which is the rotation group SO(3). The set of all orthogonal matrices of size n with determinant +1 or −1 is a representation of the (general) orthogonal group O(n). (Wikipedia).
Derivation of the rotation matrix, the matrix that rotates points in the plane by theta radians counterclockwise. Example of finding the matrix of a linear transformation Check out my Linear Equations playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmD_u31hoZ1D335sSKMvVQ90 Subs
From playlist Linear Equations
This clip gives describes a rotation matrix in 2D. The clip is from the book "Immersive Linear Algebra" available at http://www.immersivemath.com.
From playlist Chapter 6 - The Matrix
7 Rotation of reference frames
Ever wondered how to derive the rotation matrix for rotating reference frames? In this lecture I show you how to calculate new vector coordinates when rotating a reference frame (Cartesian coordinate system). In addition I look at how easy it is to do using the IPython notebook and SymPy
From playlist Life Science Math: Vectors
A point multiplied by the rotation matrix is rotated by theta degrees. What is the rotation matrix, though, and why?
From playlist Fun
Rotation matrices | Lecture 8 | Matrix Algebra for Engineers
Example of the rotation matrix as an orthogonal matrix. Join me on Coursera: https://www.coursera.org/learn/matrix-algebra-engineers Lecture notes at http://www.math.ust.hk/~machas/matrix-algebra-for-engineers.pdf Subscribe to my channel: http://www.youtube.com/user/jchasnov?sub_confir
From playlist Matrix Algebra for Engineers
Why is the Rotation Matrix Orthogonal? | Classical Mechanics
For any rotation matrix R, we usually know that it's transpose is equal to it's inverse, so that R^T R is equal to the identity matrix. This is due to the fact that we take the rotation matrix to be orthogonal. But why do we assume that rotation matrices are orthogonal? In this video, we w
From playlist Classical Mechanics
How to Calculate a Rotation Matrix | Classical Mechanics
In this video, we will show you how to calculate the rotation matrix for any given rotation. To do so, we will assume a passive rotation, that is we rotate our basis vectors. 00:00 Introduction 00:34 Example 01:25 Proof Follow us on Instagram: https://www.instagram.com/prettymuchvideo/
From playlist Classical Mechanics
Calculating the matrix of a linear transformation with respect to a basis B. Here is the case where the input basis is the same as the output basis. Check out my Vector Space playlist: https://www.youtube.com/watch?v=mU7DHh6KNzI&list=PLJb1qAQIrmmClZt_Jr192Dc_5I2J3vtYB Subscribe to my ch
From playlist Linear Transformations
Sketch a Linear Transformation of a Rectangle Given the Transformation Matrix (Reflection)
This video explains 2 ways to graph a linear transformation of a rectangle on the coordinate plane.
From playlist Matrix (Linear) Transformations
A Spectral Decomposition approach to the robust conversion of 4D Rotation matrices to double quaternions.
From playlist AGACSE2021
16: Basis Sets - Intro to Neural Computation
MIT 9.40 Introduction to Neural Computation, Spring 2018 Instructor: Michale Fee View the complete course: https://ocw.mit.edu/9-40S18 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP61I4aI5T6OaFfRK2gihjiMm Two-layer feed-forward networks, matrix transformations, basis
From playlist MIT 9.40 Introduction to Neural Computation, Spring 2018
MIT RES.TLL-004 Concept Vignettes View the complete course: http://ocw.mit.edu/RES-TLL-004F13 Instructor: Dan Frey This video leads students through describing the motion of all points on a wobbly disk as a function of time. Properties of time independent rotation matrices are explored.
From playlist MIT STEM Concept Videos
Euler Angles and the Euler Rotation Sequence
In this video we discuss how Euler angles are used to define the relative orientation of one coordinate frame to another. Topics and Timestamps: 0:00 – Introduction and example 2:34 – The Euler Rotation Sequence 16:10 – Matlab animation showing rotation sequence 21:03 – The direction cos
From playlist Flight Mechanics
17: Principal Components Analysis_ - Intro to Neural Computation
MIT 9.40 Introduction to Neural Computation, Spring 2018 Instructor: Michale Fee View the complete course: https://ocw.mit.edu/9-40S18 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP61I4aI5T6OaFfRK2gihjiMm Covers eigenvalues and eigenvectors, Gaussian distributions, co
From playlist MIT 9.40 Introduction to Neural Computation, Spring 2018
Unitary Transformations and the SVD [Matlab]
This video describes how the singular value decomposition (SVD) is related to unitary transformations, with Matlab code. Book Website: http://databookuw.com Book PDF: http://databookuw.com/databook.pdf These lectures follow Chapter 1 from: "Data-Driven Science and Engineering: Machine
From playlist Data-Driven Science and Engineering
Live Stream #148.1: 3D Rendering Basics
Drawing 3D shapes on 2D canvas. 🎥 Part 2: https://youtu.be/M_YNwb7UudI 7:53 - Matrix Multiplication 54:45 - Coding Challenge: Cube Projected on 2D Screen 🔗 Matrix Multiplication: http://matrixmultiplication.xyz 🔗 Rotation Matrix on Wikipedia: https://en.wikipedia.org/wiki/Rotation_matri
From playlist Live Stream Archive
Lecture: The Singular Value Decomposition (SVD)
Perhaps the most important concept in this course, an introduction to the SVD is given and its mathematical foundations.
From playlist Beginning Scientific Computing
Linear Algebra for Computer Scientists. 13. Transformation Matrices
Animated computer graphics are based on models composed of thousands of tiny primitive shapes such as triangles, and each vertex in a model is encoded as a vector. This computer science video demonstrates how matrices are used to transform these vector. In particular you will learn how to
From playlist Linear Algebra for Computer Scientists
[Lesson 13] QED Prerequisites - The Pauli Spin Matrices...from scratch!
The purpose of this video is to motivate the Pauli Spin matrices from first principles. We will use these matrices a lot during the study of QED and it is critical that every aspect of their design and origin is well understood. This video begins by describing what it means to "rotate a sp
From playlist QED- Prerequisite Topics
The inverse of a matrix is a similarly sized matrix such that the multiplication of the two matrices results in the identity matrix. In this video we look at an example of this. You can learn more about Mathematica on my Udemy course at https://www.udemy.com/mathematica/ PS! Wait until
From playlist Introducing linear algebra