Linear Transformations and matrix representations Reading Section 1.8 and 1.9 MyMathLab: Lesson U1.4 Learning Objectives Basic Algebraically find the image of a given vector under a linear transformation. Given a linear transformation T(x)=Ax, find x for a given b in the image of T. Determine if a vector is in the range of a given linear transformation. Use linear properties to find the image of vector under a transformation. Find the standard matrix of a linear transformation. Demonstrate understanding of the geometric interpretation of a linear transformation. Demonstrate understanding of one-to-one and onto properties. Advanced Demonstrate understanding of concepts about linear transformations and their matrices. Geometrically describe the image of a vector under a linear transformation. Prove that a transformation is linear or nonlinear. Transformation, Mappings or Functions Definitions A transformation is (or function or mapping) from to is a rule that assigns to each vector in a vector in . is called the domain of is called the codomain of The notation indicates that domain of is and that the codomain of is For any give in the domain is a vector and is called the image of The set of all images of is called the range of Key Observation Every matrix vector product is a transformation, with indicating the transformation associated with matrix A Transformation : The domain is The codomain is The range is all possible linear combinations of the columns of A, or Example: Lesson U1.4 Study Guide Thursday, May 24, 2018 10:30 AM Study Guides Page 1