Examples of change of basis, invertable functions, invertable transformations

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University of Toronto St. George
Martin, Burda

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
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