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# sol35 - Partial Solution Set Leon 3.5 3.5.2(c Let u1 =(0...

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Partial Solution Set, Leon § 3.5 3.5.2 (c) Let u 1 = (0 , 1) T and u 2 = (1 , 0) T . Then { u 1 , u 3 } is an ordered basis for R 2 . Find the transition matrix corresponding to the change of basis from the standard basis to { u 1 , u 3 } . Solution : Let U = { u 1 u 2 } . Then U is the transition matrix corresponding to the change of basis from { u 1 , u 3 } to the standard basis. It follows that U - 1 is the matrix that we’re after. Performing the computation, we find that U - 1 = 1 det ( T ) 0 - 1 - 1 0 = U , i.e., U is its own inverse. 3.5.4 Let E = { (5 , 3) T , (3 , 2) T } , and let x = (1 , 1) T , y = (1 , - 1) T , and z = (10 , 7) T . Find the values of [ x ] E , [ y ] E , and [ z ] E . Solution : From our discussion in class, we know that the transition matrix from the ordered basis E to the standard basis is just T = 5 3 3 2 . Since we want to go the other way, the transition matrix we want is W = T - 1 = 2 - 3 - 3 5 . It is now an easy step to determine [ x ] E = ( - 1 , 2) T , [ y ] E = (5 , - 8) T , and [ z ] E = ( - 1 , 5) T . 3.5.5 Let u 1 = (1 , 1 , 1) T , u 2 = (1 , 2 , 2) T , and u 3 = (2 , 3 , 4) T . Let e i denote the i th standard basis vector. In part (a) of this problem, we are to find the transition matrix corre- sponding to the change of basis from { e 1 , e 2 , e 3 } to { u 1 , u

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