{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# LA_Lecture15 - Lecture 15 Linear Transformation Eigenvalues...

This preview shows pages 1–10. Sign up to view the full content.

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 e -NTUEE SCC_01_2008

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
15- 2 Lecture 15: Linear Transformation Today Matrices for Linear Transformations Transition Matrix and Similarity Eigenvalues and Eigenvectors Reading Assignment : Secs 6.3-7.1 Final Exam 2:20 – 4:20 Scope: Sections 4.7-7.1 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 7.2-7.4
What Have You Actually Learned about LT So Far? 15- 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 one-to-one: e onto: e isomorphism(one-to-one and onto): e isomorphic space: {
15- 5 Today Matrices for Linear Transformations Transition Matrix and Similarity Eigenvalues and Eigenvectors

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 0 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 ) ( ) ( ) (

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 2 2 1 1 2 2 22 1 21 1 2 12 1 11 2 1 2 1 2 22 21 1 12 11 v
15 - 9 ) ( ) ( ) ( 2 2 1 1 2 1 2 22 12 2 1 21 11 1 n n mn n n n m m e T v e T v e T v a a a v a a a v a a a v + + + = + + + =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 62

LA_Lecture15 - Lecture 15 Linear Transformation Eigenvalues...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online