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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![Exercise 5
We want to study the span in Fn
Mat,.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 that
e(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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a4dcd82-baf4-45bc-b40b-693a3e683492%2F73a98400-4b11-4308-a805-3eb5cbf43789%2Fazd1xya_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 5
We want to study the span in Fn
Mat,.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 that
e(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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