Matrix (A
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Vector Rotation ( Rot ( (x, y (x 0, y 0 y 0 x 0 Matrix (A 3
Vector Rotation ( (x, y (x 0, y 0 y y 0 (x, y (x 0, y 0 R R x x 0 rotate by (x 0, y 0 (x, y x y = x 0 cos y 0 sin = x 0 sin + y 0 cos Matrix (A 4
Vector Rotation (3 (x, y (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
Vector Rotation (4 x = x 0 cos y 0 sin y = x 0 sin + y 0 cos (x, y (x, y (x 0, y 0 x 0 sin x 0 (x 0, y 0 y 0 y y 0 x 0 y 0 cos x 0 sin y 0 sin x y 0 sin Matrix (A 6
v = (x 0, y 0 Rot ( w = (x, y rotation operator V W v f : V W W w transformation map Matrix (A 7
Matrix v (x 0, y 0 f : V W w (x, y transformation x y = x 0 cos y 0 sin = x 0 sin + y 0 cos x ] y = cos sin ] x 0 ] sin cos y A = cos sin ] sin cos w = A w = T A ( x x x T A w Matrix (A 8
Coordinates and Coordinates Systems Rectangular Coordinate System (.5,.5 any vector (x, y coordinates (.5,.5 Non-Rectangular Coordinate System any vector 3 ( 3, 3 (x ', y ' coordinates ( 3, 3 Matrix (A 9
Standard Basis e R e e = (, 0 e = (0, R any vector (x, y standard basis {e, e } spans R e 3 R 3 e e = (, 0, 0 e = (0,, 0 e 3 = (0, 0, (x, y R 3 any vector e standard basis {e, e, e 3 } spans R 3 Matrix (A 0
Basis and Coordinates (.5,.0 =.5 e +.0 e V m basis e =.5 0] +.0 0 ] m e basis {e, e } coordinates (.5,.0 =.5.0] 0 0 ] collinear vectors linearly dependent vectors many bases but the same number of basis vectors basis {u, u } basis {v, v } basis {w, w } coordinates coordinates coordinates u (x u, y u v (x v, y v w (x w, y w v w u Matrix (A
Standard Basis & Standard Matrix e = (0, f : V W T A (e = ( sin, cos T A (e = (cos, sin e = (, 0 transformation standard basis {e, e } standard matrix A = T A (e T A (e ] x ] y = cos sin ] x 0 ] sin cos y T A ( e T A ( e T A ( e n w = w = A T A ( x x A = x T A w Matrix (A
Dimension In vector space R any one vector line R linearly independent any two non-collinear vectors plane R linearly independent any three or more vectors linearly dependent v 3 = k v + k v R R R v v v 3 v v v Matrix (A 3
Basis S = {v, v,, v n } non-empty finite set of vectors in V S S is a basis linearly independent v kv v S spans V v span(s = span{v, v,, v n } w = k v + k v R R all possible linear combination of the vectors in in S v w { w = k v + k v + + k n v n } v Matrix (A 4
inear Dependent ( {u, v, w} linearly dependent w = u + v u = w v v = w u v w v w v w u u u u + v w = 0 u + v w = 0 u + v w = 0 Matrix (A 5
inear Dependent ( {u, v, w} linearly dependent v w v w v w u u u k u + k v + k 3 w = 0 m u + m v + m 3 w = 0 n u + n v + n 3 w = 0 (k = 0 (k = 0 (k 3 = 0 (k 0 (k 0 (k 3 0 (m = 0 (m = 0 (m 3 = 0 (m 0 (m 0 (m 3 0 (n = 0 (n = 0 (n 3 = 0 (n 0 (n 0 (n 3 0 Matrix (A 6
inear Dependent (3 {v, v, v 3, v 4 } linearly dependent v v k v + k v + k 3 v 3 + k 4 v 4 = 0 (k 0 (k 0 (k 3 0 (k 4 0 v 3 v 4 0 v + m v + m 3 v 3 + m 4 v 4 = 0 (m = 0 (m 0 (m 3 0 (m 4 0 m v + m v + m 3 v 3 + m 4 v 4 = 0 (m 0 (m 0 (m 3 0 (m 4 0 Matrix (A 7
inear Dependent (4 {v, v, v 3, v 4 } linearly dependent = v v v 3 v 4 = = Matrix (A 8
inear Independent ( S = {v, v,, v n } non-empty set of vectors in V k v + k v + + k n v n = 0 the solution of the above equation v trivial solution: k = k = = k n = 0 if other solution exists if no other solution exists S linearly dependent S linearly independent v v 3 v v k v k 3 v 3 v v v v 3 v 3 v k v v Matrix (A 9
inear Independent ( S = {v, v,, v n } non-empty set of vectors in V k v + k v + + k n v n = 0 the solution of the above equation k = k = = k n = 0 v if other solution exists if no other solution exists S S linearly dependent linearly independent v at least one vector in S is a linear combination of the other vectors in S v i = k v + k v + + k i v i + k i+ 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 v + k v + + k i v i + k i+ v i+ + + k n v n S linearly independent Matrix (A 0
inear Independent (3 S = {v, v,, v n } non-empty set of vectors in V k v + k v + + k n v n = 0 the solution of the above equation v k = k = = k n = 0 if other solution exists S linearly dependent if no other solution exists S linearly independent v S = { 0 } linearly dependent S = { v } linearly independent S = { v, v } v k v linearly independent Matrix (A
Change of Basis u = (0, u' v = (k, k u = (, 0 u' Old Basis B = {u, u } B ' = {u', u' } New Basis u ' ] B = cos sin ] coordinate of u' with respect to B u' = cos u + sin u u' = sin u + cosu u ' ] B = sin cos ] coordinate of u' with respect to B v = k u' + k u' = k (cosu +sin u + k ( sin u +cosu = (k cos k sin u +(k sin +k cosu v ] B' = k k ] coordinate of v with respect to B' v ] B = k cos k sin k sin+k cos ] coordinate of v with respect to B Matrix (A
Change of Basis u = (0, u' v = (k, k u = (, 0 u' Old Basis B = {u, u } B ' = {u', u' } New Basis v ] B' = k k ] coordinate of v with respect to B' v ] B = cos sin ] k ] sin cos k coordinate of v with respect to B u ' ] B = cos sin ] coordinate of u' with respect to B v ] B = cos sin] sin cos v] B ' v ] B = P B ' B v ] B ' u ' ] B = sin cos ] coordinate of u' with respect to B P B ' B = u' ] B u' ] B ] Matrix (A 3
Transition Matrix P B ' B = u' ] B u' ] B u' n ] B ] v ] B = P B ' B v ] B ' u ' ] B u ' ] B coordinate of u' with respect to B coordinate of u' with respect to B v ] B' coordinate of v with respect to B' v ] B coordinate of v with respect to B P B B' = u ] B' u ] B' u n ] B ' ] v ] B' = P B B ' v ] B u ] B ' u ] B' coordinate of u with respect to B' coordinate of u with respect to B' v ] B coordinate of v with respect to B v ] B' coordinate of v with respect to B' Matrix (A 4
Change of Basis Example ( Rot (30 u cos sin ] sin cos u' u Rot ( 30 P B B' = u ] B' u ] B' u n ] B ' ] P B ' B = u' ] B u' ] B u' n ] B ] u' u ] B ' coordinate of u with respect to B' 3 ] u ' ] B coordinate of u' with respect to B 3 ] u ] B' coordinate of u with respect to B' 3 ] u ' ] B coordinate of u' with respect to B ] 3 P B B' = 3 ] 3 P B ' B = 3 ] 3 Matrix (A 5
Change of Basis Example ( Rot (30 u cos sin ] u' sin cos u' ( sin, cos u ' ] B (cos, sin u ' ] B u Rot ( 30 u ] B' (sin, cos u u' u cos sin ] B ' sin cos] u' (cos, sin u (cos, sin Matrix (A 6
Change of Basis Example (3 w w ' = (, u u' u u' v ] B = P B ' B v ] B ' P B ' B = u' ] B u' ] B u' n ] B ] v ] B' coordinate of v with respect to B' u ' ] B coordinate of u' with respect to B 3 ] v ] B coordinate of v with respect to B u ' ] B coordinate of u' with respect to B ] 3 w = 3 ] 3 ] = + 3 ] + 3 P B ' B = 3 ] 3 Matrix (A 7
Change of Basis Example (4 u v = (, u' v ' u u' P B B' = u ] B' u ] B' u n ] B ' ] v ] B' = P B B ' v ] B u ] B ' coordinate of u with respect to B' 3 ] v ] B coordinate of v with respect to B u ] B' coordinate of u with respect to B' 3 ] v ] B' coordinate of v with respect to B' P B B' = 3 ] 3 v ' = 3 ] 3 ] = + 3 ] + 3 Matrix (A 8
& Transition Matrix v (x 0, y 0 f : V W w (x, y the same Basis transformation the same Basis e = (0, e ' e = (, 0 transition e ' Old Basis New Basis Matrix (A 9
Projection onto the ines Through Zero ( P R line is represented by a vector a a proj a u = u a a a Matrix (A 30
Projection onto the ines Through Zero ( Finding the vector a T (e = (cos, sin R e = (, 0 vector component of e along a T (e = (sin(90, cos(90 e = (0, R 90 = (cos, sin 90 vector component of e along a Matrix (A 3
Projection onto the ines Through Zero (3 Finding the projection of the unit vectors a = a = (cos, sin P a = (cos, sin T (e e = (, 0 e = (, 0 proj a u = u a a a vector component of e along a a = cos + sin = e a = (, 0 (cos, sin = cos T (e = e a a a = (cos, cossin e a = (0, (cos, sin = sin e = (0, T (e a = (cos, sin T (e = e a a a = (cos sin, sin vector component of e along a Matrix (A 3
Projection onto the ines Through Zero (3 Finding the projection of the unit vectors a = a = (cos, sin P a = (cos, sin T (e e = (, 0 e = (, 0 proj a u = u a a a vector component of e along a a = cos + sin = e a = (, 0 (cos, sin = cos T (e = e a a a = (cos, cossin e a = (0, (cos, sin = sin e = (0, T (e a = (cos, sin T (e = e a a a = (cos sin, sin vector component of e along a Matrix (A 33
Projection onto the ines Through Zero The Standard Matrix a = (cos, sin P a = a = (cos, sin e = (, 0 T (e e = (, 0 vector component of e along a T ] = T (e T (e ] = cos sin cos sin cos sin ] = cos sin ] sin sin = P e = (0, T (e a = (cos, sin vector component of e along a Matrix (A 34
Reflections About the ines Through Zero ( H H x x line is represented by a vector a H x P x H x x P x x x x ( P x x = H x x P x x = H x ( P x I x = H x Matrix (A 35
Reflections About the ines Through Zero ( x H H x P x ( P x I x = H x P = cos sin ] sin sin P I = cos sin sin sin ] H 0 = cos sin sin cos] Matrix (A 36
References ] http://en.wikipedia.org/ ] Anton, et al., Elementary inear Algebra, 0 th ed, Wiley, 0 3] Anton, et al., Contemporary inear Algebra,