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Math 2270 Quiz 6 1. Consider an n × n matrix A. Show that there are scalars c0 , c1 , . . . , cn , not all zero, such that the matrix c0 In + c1 A + c2 A2 + · · · + cn An is noninvertible. [Hint: any n + 1 vectors from Rn will be linearly dependent.] Let ~v ∈ Rn be any nonzero vector. Then the n + 1 vectors ~v , A~v , A2~v , . . . , An~v are linearly dependent. This means that there exist constants c0 , c1 , . . . , cn such that c0~v + c1 A~v + · · · + cn An~v = ~0 ⇒ [c0 In + c1 A + · · · + cn An ]~v = ~0. This shows that ker([c0 I + c1 A + · · · + cn An ]) 6= {~0}, which is the same as saying that this matrix is not invertible. 2. Find the matrix B of the linear transformation T (~x) = A~x with respect to the basis B = {~v1 , ~v2 }, where 0 1 1 1 A= , ~v1 = , ~v2 = . 1 0 1 −1 The formula given in class and in the text for the B-matrix of a linear transformation is B = [[T (~v1 )]B · · · [T (~vn )]B ], where B = {~v1 , . . . , ~vn }. In our case, T (~vi ) = A~vi , so we compute 0 1 1 1 0 1 1 −1 A~v1 = = = ~v1 , A~v2 = = = −~v2 . 1 0 1 1 1 0 −1 1 The B coordinates are easily calculated in this case, 1 0 [A~v1 ]B = , [A~v2 ]B = . 0 −1 This gives us that B= 1 0 0 −1 .