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Lecture

Using coordinates, Algebra Transformations, Change of Basis, change of basis for transformation

7 Pages
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Department
Mathematics
Course Code
MAT224H1
Professor
Martin, Burda

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Description
Tuesday 25012011, Lecture notes by Y. Burda 1 Using coordinates We are going to expand on the idea that while one should think about vectors in vector spaces and linear transformations, computations should be done 1 with coordinates and transformation matrices . Example: Let T : P 2 be t1e dierentiation mapping T(p) = p . Let A = (1 x,1 + x,1 + x + x ), B = (1 x,x) be bases of P a2d P 1 respectively. Find [T] and use it to nd a basis for kerT and ImT. B,A Solution: To nd [T] we should apply T to basis vectors from A and B,A express the results as linear combinations of vectors from B. For the vector 1 x we have [T(1 x)]B= [1] B To nd [1] B we should express 1 as 11 x) + 2 for some 1, 2 R. If 1 = 1(1 x) + 2x, then 1 1, = 21. Thus [T(1 x)] = [1] = 1 B B Similarly we nd [T(1 + x)B = [1]B= ( 1) T(1 + x + x ) = [1 + 2x]B= ( 3 B Hence [T] = 1 1 1 B,A 1 1 3 To nd kerT we recall that v kerT if and only if its coordinates vector x = [v] solves [T] x = 0. A B,A To solve the system [T]B,A x = 0 we row-reduce the matrix 1 1 1 1 1 3 1 And remember: hours of calculations can often spare you tens of minutes of thinking! 1 www.notesolution.com
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