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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