Old & new co-ordinates

Transcription

1 Old & new co-ordinates Aim lecture: A wise choice of co-ordinates can make life much easier. We give some examples showing how to make a linear change of co-ords to facilitate calculations. Suppose we have old co-ords (x 1,..., x n ) F n & some new co-ords (y 1,..., y n ) F n. They really lie in different copies of F n & we need some notation to distinguish the two. Notn We let F n x be a copy of F n whose co-ords are (x 1,..., x n ) & F n y be another copy of F n whose co-ords are (y 1,..., y n ). We let GL n (F) denote the set of invertible n n-matrices with entries in F. Daniel Chan (UNSW) Lecture16: Change of co-ordinates Semester / 8

2 Change of co-ordinate matrix Defn A change of co-ordinates matrix (with say old co-ords (x 1,..., x n ) & new co-ords (y 1,..., y n )) is a co-ord system of the form C : F n y F n x. Note the change of co-ordinates formula (x 1,..., x n ) T = C(y 1,..., y n ) T, (y 1,..., y n ) T = C 1 (x 1,..., x n ) T Rem The possible change of co-ords matrices correspond to the elements of GL n (F). E.g. Best to visualise an example geometrically. Suppose C : R 2 y R 2 x is given by the matrix ( 3 ) C = Daniel Chan (UNSW) Lecture16: Change of co-ordinates Semester / 8

3 Reflections via change of co-ordinate system E.g. Use change of co-ordinates to find the matrix A M 22 (R) representing reflection about the line x 2 = 4 3 x 1. A We use the co-ordinate system C : R 2 y R 2 x defined by the matrix ( 3 ) C = In the new co-ords, the representing matrix is easy ( ) ( ) ( ) ( ) y1 y1 1 0 y1 =. y 2 y y 2 But our theory of matrix reprns also gives ( ) C AC = 0 1 Note Hence A = C 1 AC : R 2 y 4 5 C R 2 x 3 5 A R 2 x C 1 R 2 y. Daniel Chan (UNSW) Lecture16: Change of co-ordinates Semester / 8

4 Linear maps & change of co-ords The previous example illustrates the general principle below. Prop Let F n x be the space of old co-ords (x 1,..., x n ) & F n y be the space of new co-ords (y 1,..., y n ) which are related by the change of co-ords matrix C : F n y F n x. If A x : F n x F n x denotes a linear map in old co-ords & A y : F n y F n y denotes the corresponding map in new co-ords then A y = C 1 A x C, A x = CA y C 1 Rem It is useful to think of the composite CA y C 1 : F n x C 1 F n y A y F n y C F n x as the sequence 1 1st pass to new co-ords with C 1 2 then use linear transformation A y wrt new & easier co-ords (y 1,..., y n ). 3 finally use C to convert back to original co-ords. Daniel Chan (UNSW) Lecture16: Change of co-ordinates Semester / 8

5 Rotations via change of co-ords E.g. Write down the matrix representing rotation about the axis R v where v = 1 3 (1, 2, 2)T. Suppose the angle of rotation is θ & the direction is given by the right hand rule with thumb pointing in dirn v. A We first find a good co-ord system to use. Daniel Chan (UNSW) Lecture16: Change of co-ordinates Semester / 8

6 Rotation example cont d Note that the representing matrix in the new co-ords is Daniel Chan (UNSW) Lecture16: Change of co-ordinates Semester / 8

7 Swapping between two co-ordinate systems Let C x : F n x V, C y : F n y V be two co-ordinate systems. Then we can pass from x-co-ords to y-co-ords via F n C x C 1 y x V F n y. This suggests Defn The change of co-ordinates matrix is C = Cy 1 vector v V, its y-co-ords are y = Cx. E.g. Suppose we have new co-ord systems on R 2 given by ( ) ( ) C x =, C 3 1 y =. 1 1 C x. Given the x-co-ords x of a If the x-co-ords of v R 2 are ( x 1 x 2 ), what are the y-co-ords? What is the change of co-ords matrix? Daniel Chan (UNSW) Lecture16: Change of co-ordinates Semester / 8

8 Being careful with identifying vector spaces Isomorphisms sometimes allow us to identify two vector spaces, but this can be confusing at times, & we should be more careful as follows. Suppose given isomorphisms Φ V : V V, Φ W : W W. Defn Given isomorphisms Φ V, Φ W above, the linear map T : V W induces a corresponding linear map T = Φ 1 W T Φ V : V W. If there is no confusion in treating Φ V, Φ W as identifications, then we may also make the identification T = T. E.g. Suppose m = m m M, n = n n N so there are natural isomorphisms Φ W : F m F m1... F m M, Φ V : F n F n1... F n N given by removing internal parentheses. Given an M N-matrix of matrices defining a linear map T : F n1... F n N F m1... F m M, the corresponding linear map Φ 1 W T Φ V : F n F m is just the big matrix obtained by removing internal parentheses. Daniel Chan (UNSW) Lecture16: Change of co-ordinates Semester / 8