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 300-300
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 Rank-Nullity Theorem
- If Tx = Ax for some matrix A, then Rank(T) = Rank(A) and Nullity(T) = dim(N(A))
- Definition of One-to-One and Definition of Onto
- Theorem 6.6
- Theorem 6.7
- Theorem 6.8
- Definition of Isomorphism
- Theorem 6.9
|
Pages 312-313
2-3, 6, 10, 13-14, 19-21, 31-34, 39-42, 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. Non-standard Bases
|
Pages 322-323
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 328-329
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 350-352
11, 13, 17, 18, 19, 21, 22, 39, 40
|