Math 110: Linear Algebra Practice Final Solutions
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Question 1. (1) Write f(x) = amx m + am−1xm−1 + · · ·+ a0. Then F = f(T ) = amT + am−1Tm−1 + · · ·+a0I. Since T is upper triangular, (T )ii = T k ii for all positive k and i ∈ {1, . . . , n}. Hence Fii = amT m ii + am−1T m−1 ii + · · ·+ a0 = f(Tii). (2) TF = Tf(T ) = T (amT m+am−1Tm−1+ · · ·+a0I) = amT+am−1T+ · · ·+a0T = (amT m + am−1Tm−1 + · · ·+ a0I)T = f(T )T = FT . (3) Since T is upper triangular, so is each power of T , and hence so is F = f(T ). Therefore, Tij = 0 and Fij = 0 whenever i > j. Hence (FT )i,i+1 = n ∑
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