Change of Basis Marco Chiarandini Department of Mathematics & - PowerPoint PPT Presentation

DM559 Linear and Integer Programming Lecture 8 Change of Basis Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark

Coordinate Change Outline 1. Coordinate Change 2

Coordinate Change Resume • Linear dependence and independence • Determine linear dependency of a set of vectors, ie, find non-trivial lin. combination that equal zero • Basis • Find a basis for a linear space • Dimension (finite, infinite) 3

Coordinate Change Outline 1. Coordinate Change 4

Coordinate Change Coordinates Recall: Definition (Coordinates) If S = { v 1 , v 2 , . . . , v n } is a basis of a vector space V , then • any vector v ∈ V can be expressed uniquely as v = α 1 v 1 + · · · + α n v n • and the real numbers α 1 , α 2 , . . . , α n are the coordinates of v wrt the basis S . To denote the coordinate vector of v in the basis S we use the notation   α 1 α 2   [ v ] S = .   .   .   α n S • In the standard basis the coordinates of v are precisely the components of the vector v : v = v 1 e 1 + v 2 e 2 + · · · + v n e n • How to find coordinates of a vector v wrt another basis? 5

Coordinate Change Transition from Standard to Basis B Definition (Transition Matrix) Let B = { v 1 , v 2 , . . . , v n } be a basis of R n . The coordinates of a vector x wrt B , a = [ a 1 , a 2 , . . . , a n ] T = [ x ] B , are found by solving the linear system: a 1 v 1 + a 2 v 2 + . . . + a n v n = x that is [ v 1 v 2 · · · v n ][ x ] B = x We call P the matrix whose columns are the basis vectors: P = [ v 1 v 2 · · · v n ] Then for any vector x ∈ R n x = P [ x ] B transition matrix from B coords to standard coords moreover P is invertible (columns are a basis): [ x ] B = P − 1 x transition matrix from standard coords to B coords 6

Example           1 2 3 4    ,  , B = 2 − 1 2 [ v ] B = 1       − 1 4 1 − 5     1 2 3 2 − 1 2 P =   − 1 4 1 det ( P ) = 4 � = 0 so B is a basis of R 3 We derive the standard coordinates of v :  1   2   3   − 9   +  − 5  = 2 − 1 2 − 3 v = 4      − 1 4 1 − 5       1 2 3 4 − 9 v = 2 − 1 2 1 = − 3       − 1 4 1 − 5 − 5 B

Example (cntd)           1 2 3 5    ,  , B = 2 − 1 2  , [ x ] S = 7       − 1 4 1 − 3  We derive the B coordinates of vector x :         5 1 2 3  = a 1  + a 2  + a 3 7 2 − 1 2      − 3 − 1 4 1 either we solve P a = x in a by Gaussian elimination or we find the inverse P − 1 :   1 [ x ] B = P − 1 x = − 1 check the calculation   2 B What are the B coordinates of the basis vector? ( [ 1 , 0 , 0 ] , [ 0 , 1 , 0 ] , [ 0 , 0 , 1 ] )

Coordinate Change Change of Basis Since T ( x ) = P x then T ( e i ) = v i , ie, T maps standard basis vector to new basis vectors Example Rotate basis in R 2 by π/ 4 anticlockwise, find coordinates of a vector wrt the new basis. � 1 2 − 1 � � cos π 4 − sin π � √ √ A T = 4 = 2 1 1 sin π cos π √ √ 4 4 2 2 Since the matrix A T rotates { e 1 , e 2 } , then A T = P and its columns tell us the coordinates of the new basis and v = P [ v ] B and [ v ] B = P − 1 v . The inverse is a rotation clockwise: � � 1 1 � � � � cos ( − π 4 ) − sin ( − π 4 ) cos ( π 4 ) sin ( π 4 ) √ √ P − 1 = 2 2 = = − 1 1 sin ( − π 4 ) cos ( − π 4 ) − sin ( π 4 ) cos ( π 4 ) √ √ 2 2 9

Coordinate Change Example (cntd) Find the new coordinates of a vector x = [ 1 , 1 ] T � √ � � � 1 1 1 � � √ √ 2 [ x ] B = P − 1 x = 2 2 = − 1 1 1 0 √ √ 2 2 10

Change of basis from B to B ′ Coordinate Change Given an old basis B of R n with transition matrix P B , and a new basis B ′ with transition matrix P B ′ , how do we change from coords in the basis B to coords in the basis B ′ ? [ v ] B ′ = P − 1 v = P B [ v ] B B ′ v coordinates in B − − − − − → standard coordinates − − − − − − − → coordinates in B ′ [ v ] B ′ = P − 1 B ′ P B [ v ] B M = P − 1 B ′ P B = P − 1 . . . v n ] = [ P − 1 P − 1 P − 1 B ′ [ v 1 v 2 . . . B ′ v n ] B ′ v 1 B ′ v 2 i.e., the columns of the transition matrix M from the old basis B to the new basis B ′ are the coordinate vectors of the old basis B with respect to the new basis B ′ 11

Change of basis from B to B ′ Coordinate Change Theorem If B and B ′ are two bases of R n , with B = { v 1 , v 2 , . . . , v n } then the transition matrix from B coordinates to B ′ coordinates is given by � [ v 1 ] B ′ [ v n ] B ′ � M = [ v 2 ] B ′ · · · (i.e., the columns of the transition matrix M from the old basis B to the new basis B ′ are the coordinate vectors of the old basis B with respect to the new basis B ′ ) 12

Coordinate Change Example �� 1 � � − 1 �� �� 3 � � 5 �� B ′ = B = , , 2 1 1 2 are basis of R 2 , indeed the corresponding transition matrices from standard basis: � 1 − 1 � � 3 5 � P = Q = 2 1 1 2 have det ( P ) = 3, det ( Q ) = 1. Hence, lin. indep. vectors. We are given � 4 � [ x ] B = − 1 B find its coordinates in B ′ . 13

Example (cntd) 1. find first the standard coordinates of x � � 4 � 1 � � − 1 � � 1 − 1 � � 5 � x = 4 − = = 2 1 2 1 − 1 7 and then find B ′ coordinates: � 2 � � 5 � � − 25 � − 5 [ x ] B ′ = Q − 1 x = = − 1 3 7 16 B ′ 2. use transition matrix M from B to B ′ coordinates: [ v ] B ′ = Q − 1 P [ v ] B : v = P [ v ] B and v = Q [ v ] B ′ � � � � � � � 2 − 5 1 − 1 − 8 − 7 M = Q − 1 P = = − 1 3 2 1 5 4 � � � � � � − 8 − 7 4 − 25 [ x ] B ′ = = 5 4 − 1 16 B ′