Mathematics & Statistics Texas Tech University Kent Pearce Department of Mathematics and Statistics Texas Tech University Lubbock, Texas 79409-1042 Voice: (806)742-2566 x 226 FAX: (806)742-1112 Email: kent.pearce@ttu.edu

 Math 2360 Linear Algebra Fall 2012 Larson, Ron Linear Algebra Cengage

Review Final Exam

 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

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