# HW18MATH110S17 - HW 18 Solutions MATH 110 with Professor...

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HW 18 Solutions, MATH 110 with Professor StankovaHW 18; 3/22/2017MATH 110 Linear Algebrawith Professor Stankova7.1 #1fh (f) False. We need to take a basis ofKλwhich can be partitioned into cycles. (h) True. A pre-vious homework problem showed that for any operatorT, when dim(ker(Tk)) = dim(ker(Tk+1)),we have dim(ker(Tk)) = dim(ker(T)) for allk. Thus aftern“steps” it must stabilizesince thereVisn-dimensional.7.1 #2 (a) The characteristic polynomial isχ(t) =t2-4t+ 4, sot= 2 is the only eigenvalue.A-2I=-11-11has rank 1, so in particular we have an eigenvector (1,1)t. We want tocomptute the chain terminating here; we solve (A-2I)x= (1,1)tto findx= (0,1)t, so wehave a basisβ2={(1,1)t,(0,1)t}which is a cycle generated by (0,1)t. The Jordan normalform (in this basis) is [T]β=2102.(b) The charactersitic polynomial isχ(t)-t2-3t-4. Thus the eigenvalues aret= 4,-1.A-4Ihas kernel (2,3)tandA+Ihas kernel (1,-1)t. Thus we have a basis of eigenvectorsand in this basisβ, we have [T]β=400-1.(c) The characteristic polynomial isχ(t) =-t3+ 3t2-4.