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

Описание: 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/lin...
Free downloadable pdf of guided notes that will help you follow along with the video and use later to review what you learned.

Linear Equations in Linear Algebra:
   • Linear Algebra Full Course: Step-by-Step E...  
   • Systems of Linear Equations - Introduction...  
   • Systems of Linear Equations - Part One | L...  
   • Systems of Linear Equations - Part Two | L...  
   • Master Row Reduction and Echelon Forms | S...  
   • Master Vector Equations - Part One | Expla...  
   • Master The Concepts of Vector Independence...  
   • Master the Span of a Set of Vectors | Exam...  
   • All You Need to Know About the Matrix Equa...  
   • Applications of Linear Systems | Traffic F...  
   • Master Solutions Sets of Linear Systems | ...  
   • Difference Equations with Application Prob...  
   • Master Linear Transformations | Terminolog...  
   • Standard Matrix of a Linear Transformation...  
   • Mastering Linear Transformations in R^2: R...  
   • Mastering One-to-one And Onto Transformati...  

Matrix Algebra:
   • Master Matrix Operations: Addition, Multip...  
   • Master Finding the Inverse of a 2x2 Matrix...  
   • Master Finding the Inverse of a 3 x 3 Matr...  
   • Simplifying Matrix Expressions and Solving...  
   • Master the Invertible Matrix Theorem: Char...  
   • Master Inverse Linear Transformations: Exa...  

Determinants
   • Mastering Determinants: 2x2, 3x3, and 4x4 ...  
   • Master Properties of Determinants and Row ...  

Vector Spaces
   • Master Vector Spaces and Subspaces: Defini...  
   • Determine if a Set is a Basis for a Vector...  
   • Mastering the Null Space and Column Space ...  
   • Mastering the Dimension of a Vector Space:...  

Eigenvalues and Eigenvectors
   • Mastering Eigenvectors and  Eigenvalues | ...  
   • How to Find Eigenvalues and Eigenvectors i...  
   • Eigenvalues and The Characteristic Equatio...  
   • Diagonalizing Matrices: Eigenvalues, Eigen...  
   • Understanding Population Dynamics with Mat...  

Orthogonality and Least Squares
   • Mastering Vector Essentials: Dot Product, ...  
   • Mastering Orthogonal Projections, Decompos...  
   • Master Orthogonal Projections Onto Subspac...  
   • Mastering The Gram-Schmidt Process And QR ...  
   • Mastering Least Squares Solutions in Linea...  




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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.

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