Exercise 5We want to study the span in FnMat,.n(F) of the subset{A° | A € Matn.n(F)}of all square of matrices in Mat,n(F).We set W = Span({A³ | A € Mat,n (F)}).(a) For z E F, A E Mat,n (F), show that(zI + A) = A³ + 3zA? + 32 A + z'I,where I is the n-by-n identity matrix.(b) Let p : Mat,(F) → F be a linear functional show thate(zI + A)*) = 9(A*) + 3zp(A²) + 3z²p(A) + z*p(I).for all A E Mat,n(F) and all z E F.(c) Show that the annihilator W C (Mat,.n(F)) of W C (Mat,n(F) is equal to {0} [Hint:Use (b) for p E W](d) Show that W = Span ({A'| 4€ Mat, (F)}) = Mat,n(F).n.n

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Exercise 5 We want to study the span in F"= Matn.n(F) of the subset {A° | A E Mat,.n(F)} of all square of matrices in Matn.n(F). We set W = Span({A³ | A € Mat,nn (F)}). (a) For z E F, A E Mat,n (F), show that (zI + A) = A³ + 3zA? + 32²A+ z°I, where I is the n-by-n identity matrix. (b) Let p : Mat,.n(F) → F be a linear functional show that y(zI + A)*) = 9(A*) + 3z9(A²) + 3z²p(A) + z*p(I), for all A E Mat,.n (F) and all : E F.

Exercise 5 We want to study the span in F"= Matn.n(F) of the subset {A° | A E Mat,.n(F)} of all square of matrices in Matn.n(F). We set W = Span({A³ | A € Mat,nn (F)}). (a) For z E F, A E Mat,n (F), show that (zI + A) = A³ + 3zA? + 32²A+ z°I, where I is the n-by-n identity matrix. (b) Let p : Mat,.n(F) → F be a linear functional show that y(zI + A)) = 9(A) + 3z9(A²) + 3z²p(A) + z*p(I), for all A E Mat,.n (F) and all : E F.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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