Matrix Transformation (2A) Young Won Lim 11/9/12

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1 Matrix (A

2 Copyright (c 01 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or any later version published by the Free Softare Foundation; ith no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions to younglim@hotmail.com. This document as produced by using OpenOffice and Octave.

3 Vector Rotation (1 Rot (θ (x 1, y 1 (x 0, y 0 y 0 θ x 0 Matrix (A 3

4 Vector Rotation ( (x 1, y 1 (x 0, y 0 y 1 y 0 (x 1, y 1 (x 0, y 0 R R x 1 x 0 rotate by θ (x 0, y 0 (x 1, y 1 x 1 y 1 = x 0 cos θ y 0 sin θ = x 0 sin θ + y 0 cosθ Matrix (A 4

5 Vector Rotation (3 (x 1, y 1 (x 0, y 0 y 0 y 0 x 0 x 0 In the rotated coordinate invariant length x 0, y 0 Matrix (A 5

6 Vector Rotation (4 x 1 = x 0 cosθ y 0 sin θ y 1 = x 0 sin θ + y 0 cosθ (x 1, y 1 (x 1, y 1 (x 0, y 0 x 0 sin θ x 0 (x 0, y 0 y 0 y 1 y 0 x 0 y 0 cos θ x 0 sin θ y 0 sin θ x 1 y 0 sin θ Matrix (A 6

7 v = (x 0, y 0 Rot (θ = (x 1, y 1 rotation operator V W v f : V W W transformation map Matrix (A 7

8 Matrix v (x 0, y 0 f : V W (x 1, y 1 transformation x 1 y 1 = x 0 cos θ y 0 sin θ = x 0 sin θ + y 0 cosθ x 1 1] y = cosθ sin θ ] x 0 1] sin θ cosθ y A = cosθ sin θ ] sin θ cosθ = A = T A ( x x x T A Matrix (A 8

9 Coordinates and Coordinates Systems Rectangular Coordinate System (1.5, 1.5 any vector (x, y coordinates (1.5, 1.5 Non-Rectangular Coordinate System any vector 3 ( 3, 3 (x ', y ' coordinates ( 3, 3 Matrix (A 9

10 Standard Basis 1 e R e 1 e 1 = (1, 0 e = (0, 1 R any vector (x, y 1 standard basis {e 1, e } spans R 1 1 e 3 R 3 e e 1 = (1, 0, 0 e = (0, 1, 0 e 3 = (0, 0, 1 (x, y R 3 any vector e 1 1 standard basis {e 1, e, e 3 } spans R 3 Matrix (A 10

11 Basis and Coordinates (1.5, 1.0 = 1.5 e e V m 1 basis e = ] ] m e 1 basis {e 1, e } coordinates (1.5, 1.0 = ] ] collinear vectors linearly dependent vectors many bases but the same number of basis vectors basis {u 1, u } basis {, v } basis { 1, } coordinates coordinates coordinates u (x u, y u v (x v, y v (x, y 1 u 1 Matrix (A 11

12 Standard Basis & Standard Matrix e = (0, 1 f : V W T A (e = ( sinθ, cos θ T A (e 1 = (cos θ, sinθ e 1 = (1, 0 transformation standard basis {e 1, e } standard matrix A = T A (e 1 T A (e ] x 1 1] y = cosθ sin θ ] x 0 1] sin θ cosθ y T A ( e 1 T A ( e T A ( e n = = A T A ( x x A = x T A Matrix (A 1

13 Dimension In vector space R any one vector line R 1 linearly independent any to non-collinear vectors plane R linearly independent any three or more vectors linearly dependent v 3 = k 1 + k v R R R 1 v v v 3 Matrix (A 13

14 Basis S = {, v,, v n } non-empty finite set of vectors in V S S is a basis linearly independent kv v S spans V span(s = span{, v,, v n } = k 1 + k v R R all possible linear combination of the vectors in in S v { = k 1 + k v + + k n v n } Matrix (A 14

15 Linear Dependent (1 {u, v, } linearly dependent = u + v u = v v = u v v v u u u u + v = 0 u + v = 0 u + v = 0 Matrix (A 15

16 Linear Dependent ( {u, v, } linearly dependent v v v u u u k 1 u + k v + k 3 = 0 m 1 u + m v + m 3 = 0 n 1 u + n v + n 3 = 0 (k 1 = 0 (k = 0 (k 3 = 0 (k 1 0 (k 0 (k 3 0 (m 1 = 0 (m = 0 (m 3 = 0 (m 1 0 (m 0 (m 3 0 (n 1 = 0 (n = 0 (n 3 = 0 (n 1 0 (n 0 (n 3 0 Matrix (A 16

17 Linear Dependent (3 {, v, v 3, v 4 } linearly dependent v k 1 + k v + k 3 v 3 + k 4 v 4 = 0 (k 1 0 (k 0 (k 3 0 (k 4 0 v 3 v m v + m 3 v 3 + m 4 v 4 = 0 (m 1 = 0 (m 0 (m 3 0 (m 4 0 m 1 + m v + m 3 v 3 + m 4 v 4 = 0 (m 1 0 (m 0 (m 3 0 (m 4 0 Matrix (A 17

18 Linear Dependent (4 {, v, v 3, v 4 } linearly dependent = v v 3 v 4 = = Matrix (A 18

19 Linear Independent (1 S = {, v,, v n } non-empty set of vectors in V k 1 + k v + + k n v n = 0 the solution of the above equation v trivial solution: k 1 = k = = k n = 0 if other solution exists if no other solution exists S linearly dependent S linearly independent v 3 v v k v k 3 v 3 v v v 3 v 3 k 1 Matrix (A 19

20 Linear Independent ( S = {, v,, v n } non-empty set of vectors in V k 1 + k v + + k n v n = 0 the solution of the above equation k 1 = k = = k n = 0 v if other solution exists if no other solution exists S S linearly dependent linearly independent at least one vector in S is a linear combination of the other vectors in S v i = k 1 + k v + + k i 1 v i 1 + k i+1 v i k n v n S linearly dependent no vector in S is a linear combination of the other vectors in S v i = k 1 + k v + + k i 1 v i 1 + k i+1 v i k n v n S linearly independent Matrix (A 0

21 Linear Independent (3 S = {, v,, v n } non-empty set of vectors in V k 1 + k v + + k n v n = 0 the solution of the above equation v k 1 = k = = k n = 0 if other solution exists S linearly dependent if no other solution exists S linearly independent S = { 0 } linearly dependent S = { } linearly independent S = {, v } k v linearly independent Matrix (A 1

22 Change of Basis u = (0, 1 u' v = (k 1, k u 1 = (1, 0 u' 1 Old Basis B = {u 1, u } B ' = {u' 1, u' } Ne Basis u ' 1 ] B = cosθ sin θ ] coordinate of u' 1 u' 1 = cosθ u 1 + sin θu u' = sin θu 1 + cosθu u ' ] B = sin θ cosθ ] coordinate of u' v = k 1 u' 1 + k u' = k 1 (cosθu 1 +sin θ u + k ( sin θ u 1 +cosθu = (k 1 cosθ k sin θu 1 +(k 1 sin θ+k cosθu v ] B' = k 1 k ] coordinate of v ' v ] B = k 1cos θ k sinθ k 1 sinθ+k cos θ ] coordinate of v Matrix (A

23 Change of Basis u = (0, 1 u' v = (k 1, k u 1 = (1, 0 u' 1 Old Basis B = {u 1, u } B ' = {u' 1, u' } Ne Basis v ] B' = k 1 k ] coordinate of v ' v ] B = cosθ sin θ] k 1 ] sinθ cos θ k coordinate of v u ' 1 ] B = cosθ sin θ ] coordinate of u' 1 v ] B = cos θ sinθ] sin θ cosθ v] B ' v ] B = P B ' B v ] B ' u ' ] B = sin θ cosθ ] coordinate of u' P B ' B = u' 1 ] B u' ] B ] Matrix (A 3

24 Transition Matrix P B ' B = u' 1 ] B u' ] B u' n ] B ] v ] B = P B ' B v ] B ' u ' 1 ] B u ' ] B coordinate of u' 1 coordinate of u' v ] B' coordinate of v ' v ] B coordinate of v P B B' = u 1 ] B' u ] B' u n ] B ' ] v ] B' = P B B ' v ] B u 1 ] B ' u ] B' coordinate of u 1 ' coordinate of u ' v ] B coordinate of v v ] B' coordinate of v ' Matrix (A 4

25 Change of Basis Example (1 Rot (30 u cosθ sin θ] sin θ cosθ u' u 1 Rot ( 30 P B B' = u 1 ] B' u ] B' u n ] B ' ] P B ' B = u' 1 ] B u' ] B u' n ] B ] u' 1 u 1 ] B ' coordinate of u 1 ' ] 1 u ' 1 ] B coordinate of u' 1 ] 1 u ] B' coordinate of u ' 1 ] u ' ] B coordinate of u' 1 ] P B B' = 1 1 ] P B ' B = 1 ] 1 Matrix (A 5

26 Change of Basis Example ( ' = (1,1 u u' u 1 u' 1 v ] B = P B ' B v ] B ' P B ' B = u' 1 ] B u' ] B u' n ] B ] v ] B' coordinate of v ' u ' 1 ] B coordinate of u' 1 ] 1 v ] B coordinate of v u ' ] B coordinate of u' 1 ] = 1 ] 1 1 1] = 1+ ] 1+ P B ' B = 1 ] 1 Matrix (A 6

27 Change of Basis Example (3 u v = (1,1 u' v ' u 1 u' 1 P B B' = u 1 ] B' u ] B' u n ] B ' ] v ] B' = P B B ' v ] B u 1 ] B ' coordinate of u 1 ' ] 1 v ] B coordinate of v u ] B' coordinate of u ' 1 ] v ] B' coordinate of v ' P B B' = 1 1 ] v ' = 1 1 ] 1 1] = 1+ ] 1+ Matrix (A 7

28 & Transition Matrix v (x 0, y 0 f : V W (x 1, y 1 the same Basis transformation the same Basis e = (0, 1 e ' e 1 = (1, 0 transition e ' 1 Old Basis Ne Basis Matrix (A 8

29 Projection onto the Lines Through Zero (1 L P θ L θ θ e = (0, 1 T (e 1 = (cos θ, sin θ L R θ L θ e 1 = (1, 0 θ vector component of u along a proj a u = u a a a Matrix (A 9

30 Projection onto the Lines Through Zero a = (cosθ, sin θ L R θ a = (cosθ, sin θ L θ e 1 = (1, 0 θ olor T ( e 1 = (1, 0 e 1 = (1, 0 vector component of u along a Matrix (A 30 proj a u = u a a a

31 References 1] ] 3] M.L. Boas, Mathematical Methods in the Physical Sciences 4] D.G. Zill, Advanced Engineering Mathematics

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