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Change of Basis DM559 Linear and Integer Programming Lecture 8 Change of Basis Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline Coordinate Change 1. Coordinate Change 2 Resume Coordinate Change • 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 Outline Coordinate Change 1. Coordinate Change 4 Coordinates Coordinate Change Recall: Definition (Coordinates) If S = fv1; v2;:::; vng is a basis of a vector space V , then • any vector v 2 V can be expressed uniquely as v = α1v1 + ··· + αnvn • and the real numbers α1; α2; : : : ; αn are the coordinates of v wrt the basis S. To denote the coordinate vector of v in the basis S we use the notation 2 3 α1 6α27 [v] = 6 7 S 6 . 7 4 . 5 α n S • In the standard basis the coordinates of v are precisely the components of the vector v: v = v1e1 + v2e2 + ··· + vnen • How to find coordinates of a vector v wrt another basis? 5 Transition from Standard to Basis B Coordinate Change Definition (Transition Matrix) n Let B = fv1; v2;:::; vng be a basis of R . The coordinates of a vector x wrt B, T a = [a1; a2;:::; an] = [x]B , are found by solving the linear system: a1v1 + a2v2 + ::: + anvn = x that is [v1 v2 ··· vn][x]B = x We call P the matrix whose columns are the basis vectors: P = [v1 v2 ··· vn] Then for any vector x 2 Rn x = P[x]B transition matrix from B coords to standard coords moreover P is invertible (columns are a basis): −1 [x]B = P x transition matrix from standard coords to B coords 6 Example 82 3 2 3 2 39 2 3 < 1 2 3 = 4 B = 4 2 5 ; 4−15 ; 425 [v]B = 4 1 5 : −1 4 1 ; −5 2 1 2 33 P = 4 2 −1 25 −1 4 1 det(P) = 4 6= 0 so B is a basis of R3 We derive the standard coordinates of v: 2 1 3 2 2 3 233 2−93 v = 4 4 2 5 + 4−15 − 5 425 = 4−35 −1 4 1 −5 2 1 2 33 2 4 3 2−93 v = 4 2 −1 25 4 1 5 = 4−35 −1 4 1 −5 −5 B Example (cntd) 82 3 2 3 2 39 2 3 < 1 2 3 = 5 B = 4 2 5 ; 4−15 ; 425 ; [x]S = 4 7 5 : −1 4 1 ; −3 We derive the B coordinates of vector x: 2 5 3 2 1 3 2 2 3 233 4 7 5 = a1 4 2 5 + a2 4−15 + a3 425 −3 −1 4 1 either we solve Pa = x in a by Gaussian elimination or we find the inverse P−1: 2 1 3 −1 [x]B = P x = 4−15 check the calculation 2 B What are the B coordinates of the basis vector? ([1; 0; 0]; [0; 1; 0]; [0; 0; 1]) Change of Basis Coordinate Change Since T (x) = Px then T (ei ) = vi , ie, T maps standard basis vector to new basis vectors Example Rotate basis in R2 by π=4 anticlockwise, find coordinates of a vector wrt the new basis. " # cos π − sin π p1 − p1 A = 4 4 = 2 2 T sin π cos π p1 p1 4 4 2 2 Since the matrix AT rotates fe1; e2g, then AT = P and its columns tell us the coordinates of the −1 new basis and v = P[v]B and [v]B = P v. The inverse is a rotation clockwise: " # cos(− π ) − sin(− π ) cos( π ) sin( π ) p1 p1 P−1 = 4 4 = 4 4 = 2 2 sin(− π ) cos(− π ) − sin( π ) cos( π ) − p1 p1 4 4 4 4 2 2 9 Coordinate Change Example (cntd) Find the new coordinates of a vector x = [1; 1]T " 1 1 # p p p 1 2 [x] = P−1x = 2 2 = B − p1 p1 1 0 2 2 10 Change of basis from B to B0 Coordinate Change n Given an old basis B of R with transition matrix PB , 0 and a new basis B with transition matrix PB0 , how do we change from coords in the basis B to coords in the basis B0? −1 v=P [v] [v] 0 =P 0 v coordinates in B −−−B −B−!standard coordinates −−B −−B−−−coordinates! in B0 −1 [v]B0 = PB0 PB [v]B −1 −1 −1 −1 −1 M = PB0 PB = PB0 [v1 v2 ::: vn] = [PB0 v1 PB0 v2 ::: PB0 vn] i.e., the columns of the transition matrix M from the old basis B to the new basis B0 are the coordinate vectors of the old basis B with respect to the new basis B0 11 Change of basis from B to B0 Coordinate Change Theorem If B and B0 are two bases of Rn, with B = fv1; v2;:::; vng then the transition matrix from B coordinates to B0 coordinates is given by M = [v1]B0 [v2]B0 ··· [vn]B0 (i.e., the columns of the transition matrix M from the old basis B to the new basis B0 are the coordinate vectors of the old basis B with respect to the new basis B0) 12 Coordinate Change Example 1 −1 3 5 B = ; B0 = ; 2 1 1 2 are basis of R2, 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 B0. 13 Example (cntd) 1. find first the standard coordinates of x 1 −1 1 −1 4 5 x = 4 − = = 2 1 2 1 −1 7 and then find B0 coordinates: −1 2 −5 5 −25 [x] 0 = Q x = = B −1 3 7 16 B0 2. use transition matrix M from B to B0 coordinates: −1 v = P[v]B and v = Q[v]B0 [v]B0 = Q P[v]B : 2 −5 1 −1 −8 −7 M = Q−1P = = −1 3 2 1 5 4 −8 −7 4 −25 [x] 0 = = B 5 4 −1 16 B0.
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