(a) Let f: V > V be any linear map where V is a vector space of dimension n over a field K. Suppose there is a vector v E V so that v, f(v),... ,f"-1 (v) are independent, hence form a basis of V. (a.1 Show that there exist a, ai, ..., an-1 E K so that f"(v)= an-1f"(v) + ... + af(v) + a0v. (a.2) Use (a.1) to find the matrix of f under the basis {v,f(v),... ,f"-1 (v)} (a.3) Find the characteristic polynomial of f
(a) Let f: V > V be any linear map where V is a vector space of dimension n over a field K. Suppose there is a vector v E V so that v, f(v),... ,f"-1 (v) are independent, hence form a basis of V. (a.1 Show that there exist a, ai, ..., an-1 E K so that f"(v)= an-1f"(v) + ... + af(v) + a0v. (a.2) Use (a.1) to find the matrix of f under the basis {v,f(v),... ,f"-1 (v)} (a.3) Find the characteristic polynomial of 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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![(a) Let f: V > V be any linear map where V is a vector space of dimension n over a
field K. Suppose there is a vector v E V so that v, f(v),... ,f"-1 (v) are independent,
hence form a basis of V.
(a.1 Show that there exist a, ai, ..., an-1 E K so that
f"(v)= an-1f"(v) + ... + af(v) + a0v.
(a.2) Use (a.1) to find the matrix of f under the basis {v,f(v),... ,f"-1 (v)}
(a.3) Find the characteristic polynomial of f](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0282592e-adf8-40c1-9d31-7bca34586c75%2Fbd4cfc44-5ade-4c76-a7d1-31744c2ea3d6%2Fy139889.png&w=3840&q=75)
Transcribed Image Text:(a) Let f: V > V be any linear map where V is a vector space of dimension n over a
field K. Suppose there is a vector v E V so that v, f(v),... ,f"-1 (v) are independent,
hence form a basis of V.
(a.1 Show that there exist a, ai, ..., an-1 E K so that
f"(v)= an-1f"(v) + ... + af(v) + a0v.
(a.2) Use (a.1) to find the matrix of f under the basis {v,f(v),... ,f"-1 (v)}
(a.3) Find the characteristic polynomial of f
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