Study Guides
(238,472)

Canada
(115,151)

University of Toronto St. George
(7,977)

Mathematics
(538)

MAT224H1
(48)

Martin, Burda
(2)

Final

# Examples of change of basis, invertable functions, invertable transformations

by
OneClass6291

Unlock Document

University of Toronto St. George

Mathematics

MAT224H1

Martin, Burda

Winter

Description

Wednesday 26012011, Lecture notes by Y. Burda 1 Example of Changing Bases Last time weve seen the formulae [v]A = [I] A ,A[v]A [T] B ,A= [I]B ,B [T]B,A [IA,A In particular changing the basis in the domain of T results in performing column-operations to its transformation matrix. Similarly, changing the basis in the range of T results in performing row-operations to its transformation matrix. Example: Find bases A , B of P and P so2that the 1ransformation matrix [T] B ,A is in its reduced row-echelon form, where T : P 2 P , 1 T(a + bx + cx ) = (a 3b + c) + (2a 6b + 3c)x The idea is to start with arbitrary bases A,B of P and P . With respect 2 1 to these bases the matrix of the transformation T is likely not to be in its row-reduced form. We can row-reduce it and nd a matrix P so that P [T] B,A is the row-reduced form of [T] . We can then try to nd a basis B such B,A that [I]B ,B = P. Once such a basis is found we will get [T] B ,A = [I]B ,B[T] B,A is in the reduced row-echelon form. Lets carry out this plan now: 2 Let A = (1,x,x ), B = (1,x) be the standard bases of P and P . 2 1 To nd [T] we compute T(1) = 1+2x,T(x) = 36x,T(x ) = 1+3x 2 B,A and put the coecients as columns of the transformation matrix: 1 3 1 [T]B,A = 2 6 3 Now we should reduce it using row operations and also make a record of the operations we are doing to nd the matrix P such that P [T] is in B,A the row-reduced form. We can nd P in a simpler way: multiplying any matrix by P from the left is equivalent to performing on this matrix the row operations that we do to bring [T] to its row-reduced form. In particular B,A if we perform these operations on the identity matrix, we will get P. 1 www.notesolution.com

More
Less
Related notes for MAT224H1