. In this exercise we prove that the columns of P in Theorem6.18 are linearly
Chapter 6, Problem 50(choose chapter or problem)
. In this exercise we prove that the columns of P in Theorem6.18 are linearly independent and therefore P is invertible.The proof is by contradiction: Suppose that Re(u) and Im(u)are linearly dependent. Then there exists a real scalar c such thatRe(u) = cIm(u).(a) Prove thatu = Re(u) + iIm(u) = (c + i)Im(u)and from this show u = (c + i)Im(u).(b) Show that Re(u) = cIm(u) by evaluating ARe(u)andAcIm(u) and setting the results equal to each other. (HINT:Use (a) from Exercise 49 and that u is an eigenvector witheigenvalue .)(c) Show that u = Re(u) + iIm(u), and combine this withthe result from (b) to prove that u = (c + i)Im(u).(d) Prove that (c + i)Im(u) = (c + i)Im(u). Show thatIm(u) = Im(u), and explain why this implies is a realnumber.(e) Explain why being a real number is a contradiction, andfrom this complete the proof.
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