Mastering Linear Transformations in R^2: Reflection, Rotation, Shear & More

In this video, we dive deep into geometric linear transformations in R^2 as part of our Linear Algebra series. You'll learn how to find the standard matrix for various transformations such as reflection, projection, rotation, contraction, expansion, and shear. Through clear examples, we explore how these transformations map vectors in the Euclidean plane, all while preserving linearity. 🔑 Key Topics: Reflection across axes and the origin Rotation transformations using trigonometric functions Projection onto axes Contraction and expansion transformations Shear transformations Don't forget to check out the free guided notes linked below to follow along with this lesson and reinforce your understanding! Whether you're a beginner or preparing for a linear algebra exam, this video has you covered. *Link to Guided Notes* https://www.understandthemath.com/linear-algebra Free downloadable pdf of guided notes that will help you follow along with the video and use later to review what you learned. _Example Problem Review Videos:_ https://youtu.be/bVd_x5l9Zfc https://youtu.be/ASr-SFz6n1o https://youtu.be/aplIzmHw-u4 https://youtu.be/__QZVfNZ6HQ https://youtu.be/O2bFill56es https://youtu.be/9veowVi9Jzc https://youtu.be/jRZnGm9TXqY _Linear Equations in Linear Algebra:_ https://youtu.be/_l43mbQICmo https://youtu.be/sI1FGsOFWRU https://youtu.be/35uQZLJlvkw https://youtu.be/LyWv0J6yBjY https://youtu.be/-Uz-0ARXcTI https://youtu.be/9DYGh3-rxKw https://youtu.be/a4x_DbEqftY https://youtu.be/W59Tgy4FknY https://youtu.be/EhPzJ1d-lIU https://youtu.be/IfbdK1bjBXE https://youtu.be/_SqHWIMNBR4 https://youtu.be/GIsQYQRd-G4 https://youtu.be/oo80p9HhLrY https://youtu.be/e7V7aXIwaYk https://youtu.be/6AqORKUI6KI https://youtu.be/VoGvUpr3jB4 _Matrix Algebra:_ https://youtu.be/XeG9NPn9VgQ https://youtu.be/SalKbfqRJuY https://youtu.be/ZmMNp3Je73A https://youtu.be/fphchnlCx70 https://youtu.be/MSMyWUDwIJ4 https://youtu.be/0lBavPQk7uo _Determinants_ https://youtu.be/ajM4FRgi1LQ https://youtu.be/nO5F-Qtn1wE _Vector Spaces_ https://youtu.be/Niz4RbwxOfw https://youtu.be/fhzuPsEi9KY https://youtu.be/2lMyB5sgSKY https://youtu.be/Nhcu9iHLuV4 _Eigenvalues and Eigenvectors_ https://youtu.be/UwZdVMd5wv0 https://youtu.be/ZHKujIVw0BM https://youtu.be/A3do1ewR6ZA https://youtu.be/SGRNgOWh5Kc https://youtu.be/-SxY7fe9xKw _Orthogonality and Least Squares_ https://youtu.be/rLLmyfXNh38 https://youtu.be/U4CSYB1nm2w https://youtu.be/kEtnqHVtVok https://youtu.be/na7tk-yc7I0 https://youtu.be/F7AFmU1tzvk https://youtu.be/f8DVtPvqtwc *Subscribe to My YouTube Channel* https://www.youtube.com/channel/UCeHFoyFD9WSHlPVdkB0g8RA?sub_confirmation=1 *Ordinary Differential Equations Playlist* https://www.youtube.com/playlist?list=PL_87SXcKvr8W0EFz5p0C3Y8efuGxzCgH8 *Visit the Understand The Math Website* https://www.understandthemath.com Find free guided notes, filled in guided notes for purchase, course outlines/playlists, and math merchandise. *Connect & Support* TikTok: https://www.tiktok.com/@understandthemath Instagram: https://www.instagram.com/understandthemath Facebook: https://www.facebook.com/profile.php?id=61552291723643&mibextid=LQQJ4d Support: https://buymeacoffee.com/understandthemath *Timestamps* 00:00 – Introduction to Linear Transformations in R^2 00:31 – Reflection Transformations Overview 01:58 – Reflection Across the X-Axis 03:32 – Reflection Across the Y-Axis 05:06 – Reflection Across the Line Y=X 06:22 – Reflection About the Origin 07:35 – Example: Combining Two Reflection Transformations 09:40 – Projection Transformations Overview 10:12 – Projection Onto the X-Axis 11:10 – Projection Onto the Y-Axis 12:14 – Rotation Transformations Overview 13:45 – Example: Counterclockwise Rotation by pi/6 15:15 – Example: Combining Reflection and Rotation Transformations 16:40 – Contraction and Expansion Transformations Overview 18:50 – Example: Horizontal Expansion and Vertical Contraction 20:14 – Shear Transformations Overview 21:30 – Horizontal Shear Transformation 23:15 – Vertical Shear Transformation 24:10 – Example: Horizontal Shear with Unchanged *More About Understand The Math* Understand The Math offers clear, accessible, and engaging mathematics instruction for students and lifelong learners. I'm Cheryl Hile, the founder of Understand The Math, with a Ph.D. in Engineering Science and Applied Mathematics and over 25 years of university teaching experience. My goal is to provide accurate, high-quality math instruction through step-by-step explanations and carefully worked-out examples. Each video includes free guided notes, linked in the description. These notes help students follow along, highlight key concepts, and practice problems, making review easier and more effective. #LinearAlgebra #LinearTransformations #UnderstandTheMath #Reflection #Projection #Rotation #Contraction #Expansion #Shear #MatrixMath #GeometricTransformations #LearnMath #MathTutorial