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 Summer II 2008 Leon, Steven J Linear Algebra Pearson

Review Exam III
 Section Content Suggested Problems Section 3.5 Rn Standard Basis for Rn Ordered Basis in Rn Coordinates w.r.t Ordered Basis Problem 1. Given coordinates in standard basis find coordinates in ordered basis 2. Given coordinates in ordered basis find coordinates in standard basis Solution Problem 2: Transition Matrix from Ordered Basis to Standard Basis U Problem 1: Transition Matrix from Standard Basis to Ordered Basis U-1 Transition Matrix S from Ordered Basis F to Ordered Basis E General Finite Dimensional Vector Space Ordered Basis Coordinates w.r.t Ordered Basis Problem Given coordinates in ordered basis E find coordinates in ordered basis F Solution Problem 2: Transition Matrix S from Ordered Basis F to Ordered Basis E 1, 2, 3, 5, 6, 10 Section 3.6 vector space (linear space) Row Space of A Column Space of A Theorem: Two row equivalent matrices have the same row space Theorem 3.6.2 Consistency Theorem for Linear Systems Theorem: A (nxn) is non-singular iff the column space of A = Rn Rank of A Nullity of A Theorem 3.6.5. Rank-Nullity Theorem Meta-Theorem: Two row equivalent matrices have different column spaces, but the same column dependency relationships Theorem 3.6.6 dim(row space of A) = dim(col space of A) 1, 2, 4, 7, 8, 10, 11, 12 Section 4.1 Examples Matrix Multiplication Properties L(0V) = 0W L(-x) = -L(x) Linear combinations Kernel of L Image of a Subspace under L Range of L Theorem 4.1.1 1, 4, 5, 6, 7, 9, 11, 17, 19 Section 4.2 Theorem 4.2.1 Matrix representation of Linear Transformation from Rn to Rm Theorem 4.2.2 Matrix representation of Linear Transformation from V to W Theorem 4.3.3 Construction of Matrix A for case of V = Rn and W = Rm in Theorem 4.2.2 Corollary Above construction via row equivalent transformation of an augmented matrix 1, 2, 3, 4, 6, 14, 18 Section 4.3 Linear Operator Theorem 4.3.1 Similarity 1, 3, 4, 5

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