Title Linear Transformation_01.pdf ✅

Page 1 of 16 MAT 216 Linear Algebra & Fourier Analysis Contents:  Introduction to Linear Transformation & Visualization  Matrix form of Linear Transformation  Linear Transformation as Matrix Vector Multiplication  Loss of dimension in Linear transformation Reference Book: Elementary Linear Algebra, Larson | Edwards | Falvo | 6th Ed.
MAT 216 Lecture 3 Page 2 of 16 Linear Transformation Transformation: Transformation is actually mapping each element of the domain into some element of the codomain. It acts like a function. Definition: A transformation is a function that maps a set to itself. Example: Linear Transformation: A linear transformation is a function that maps a vector space into another vector space : Domain of Codomain of Two Properties of Linear Transformation: 1. , 2. , The elements of the set of are and . The elements of the set of are and We can combine the two properties into a single one: ; Linear transformations preserve linear combination.
MAT 216 Lecture 3 Page 3 of 16 Consider few examples: 1. Verify a linear transformation from into or ( ) ( ) Proof: Let ( ), ( ) Addition law: ( ) ( ) ( ) ( ) ( ) Since ( ) ( ) ( ) ( ) ( ) ( ) ( ) Scalar Multiplication law: ( ) ( ) ( ) ( ) ( ) ( ) is a linear transformation.