Section

Content

Suggested Problems

Section 6.1

 Terminology
 Map, Function, Transformation
 Domain, Codomain, Image, Range, Preimage
 Definition of Linear Transformation
 Examples
 Linear Transformations defined bia Matrix Multiplication
 Theorem 6.1 Properties of Linear Transformations
 Linear Transformation Determined by Its Action on a Basis

Pages 300300
3, 6, 10, 11, 14, 16, 21, 22, 25, 27, 2934, 37

Section 6.2

 Kernel of a Linear Transformation T
 Theorem 6.3 Ker(T) is a Subspace of the Domain of T
 If Tx = Ax for some matrix A, then Ker(T) = N(A)
 Range of Linear Transformation T
 Theorem 6.4 Range of T is a Subspace of the Codomain of T
 If Tx = Ax for some matrix A, then Ker(T) = Column Space of A
 Definition of Rank and Nullity of a Linear Transformation T
 Theorem 6.5 RankNullity Theorem
 If Tx = Ax for some matrix A, then Rank(T) = Rank(A) and Nullity(T) = dim(N(A))
 Definition of OnetoOne and Definition of Onto
 Theorem 6.6
 Theorem 6.7
 Theorem 6.8
 Definition of Isomorphism
 Theorem 6.9

Pages 312313
23, 6, 10, 1314, 1921, 3134, 3942, 52

Section 6.3

 Standard Matrix for a Linear Transformation
 Composition of Linear Transformations
 Standard Matrix for a Composition is given by Matrix Multiplication of the Matrices of Component Transformations
 Definition of an Inverse Linear Transformation
 Existence of an Inverse Transformation
 Transformation Matrix w.r.t. Nonstandard Bases

Pages 322323
1, 3, 5, 8, 10, 25, 26, 37, 38

Section 6.4

 Finding the Matrix for Linear Transformation
 Definition of Similar Matrices
 Theorem 6.13 Properties of Similar matrices

Pages 328329
1, 3, 5, 7

Section 7.1

 Eigenvalue Problem
 Definition of Eigenvalues and Eigenvectors
 Theorem 7.1 Eigenspace of a Matrix forms a Subspace
 Characteristic Polynomial of a Matrix
 Procedure for Finding Eigenvalues and Eigenvectors of a Matrix
 Eigenvalues of Triangular Matrices

Pages 350352
11, 13, 17, 18, 19, 21, 22, 39, 40
