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Unformatted text preview: Lecture 15 Linear Transformation & Eigenvalues and Eigenvectors Last Time Introduction to Linear Transformations The Kernel and Range of a Linear Transformation Elementary Linear Algebra R. Larsen et al. (5 Edition) TKUEE eNTUEE SCC_01_2008 15 2 Lecture 15: Linear Transformation Today Matrices for Linear Transformations Transition Matrix and Similarity Eigenvalues and Eigenvectors Reading Assignment : Secs 6.37.1 Final Exam 2:20 4:20 Scope: Sections 4.77.1 70% Sections 1.1 4.6 30% Tip: Practice your homework problems and really understand Makeup Lecture Diagonalization Symmetric Matrices and Orthogonal Diagonalization Applications Reading Assignment : Secs 7.27.4 What Have You Actually Learned about LT So Far? 15 3 15  4 Keywords in Section 6.2: kernel of a linear transformation T : e T a range of a linear transformation T : e T a rank of a linear transformation T : e T a nullity of a linear transformation T : e T a onetoone: e onto: e isomorphism(onetoone and onto): e isomorphic space: { 15 5 Today Matrices for Linear Transformations Transition Matrix and Similarity Eigenvalues and Eigenvectors 15  6 6.3 Matrices for Linear Transformations ) 4 3 , 2 3 , 2 ( ) , , ( ) 1 ( 3 2 3 2 1 3 2 1 3 2 1 x x x x x x x x x x x T + + + = Three reasons for matrix representation of a linear transformation:  = = 3 2 1 4 3 2 3 1 1 1 2 ) ( ) 2 ( x x x A T x x It is simpler to write. It is simpler to read. It is more easily adapted for computer use. Two representations of the linear transformation T : R 3 R 3 : 15  7 Thm 6.10: (Standard matrix for a linear transformation) such that on ansformati linear trt a be : Let m n R R T , ) ( , , ) ( , ) ( 2 1 2 22 12 2 1 21 11 1 = = = mn n n n m m a a a e T a a a e T a a a e T ) ( to correspond columns n se matrix who Then the i e T n m . for matrix standard the called is A . in every for ) ( such that is T R A T n v v v = [ ] = = mn m m n n n a a a a a a a a a e T e T e T A 2 1 2 22 21 1 12 11 2 1 ) ( ) ( ) ( 15  8 Pf: n n n e v e v e v v v v + + + = = 2 2 1 1 2 1 v ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( L.T. a is 2 2 1 1 2 2 1 1 2 2 1 1 n n n n n n e T v e T v e T v e v T e v T e v T e v e v e v T T T + + + = + + + = + + + = v + + + + + + + + + = = n mn m m n n n n n mn m m n n v a v a v a v a v a v a v a v a v a v v v a a a a a a a a a A...
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This note was uploaded on 07/22/2011 for the course ME 3 taught by Professor Prof.ramachandran during the Spring '11 term at Indian Institute of Technology, Kharagpur.
 Spring '11
 Prof.Ramachandran

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