(a) Let f: V > V be any linear map where V is a vector space of dimension n over afield 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 thatf"(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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Asked Oct 25, 2019
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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
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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

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Expert Answer

Step 1

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 ∈ V so that v, f(v), … ,fn-1(v) are independent, hence for a basis.

For (a. 1)
By the property of basis, we know that other than basis elements of the set can
be written as linear combination of the basis elements
Hence, f"(v) can be written as the linear combination of basis elements.
In other words
f"(v) an-1f"-1(v) + .-. + af(v) + aqv ----
such that at least one of an-1, ... ,a1,ag E K is non-zero
--(1)
So, (a. 1) is justified.
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For (a. 1) By the property of basis, we know that other than basis elements of the set can be written as linear combination of the basis elements Hence, f"(v) can be written as the linear combination of basis elements. In other words f"(v) an-1f"-1(v) + .-. + af(v) + aqv ---- such that at least one of an-1, ... ,a1,ag E K is non-zero --(1) So, (a. 1) is justified.

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Step 2

For (a. 2),

We will use (a. 1) to find the matrix of f under the basis {v, f(v), .. ,f"-1(v)}.
f(v)
So,
aov
f(f(v))= f2(v) = aif (v) + agv
f(fn-1(v)) fn (v) = an-1f"-1(v)
a1f(v)aov
So, the matrix is formed as
аo
а.
а
0
а
а
(say) A
0 0
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We will use (a. 1) to find the matrix of f under the basis {v, f(v), .. ,f"-1(v)}. f(v) So, aov f(f(v))= f2(v) = aif (v) + agv f(fn-1(v)) fn (v) = an-1f"-1(v) a1f(v)aov So, the matrix is formed as аo а. а 0 а а (say) A 0 0

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Step 3

For (a. 3),

...
Now, we need to find the characteristic polynomial of f.
аo
аo
0
а,
Char( det(A-x1 =det
0
0
3 (а, — х)(а - х) -(а,
— х)
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Now, we need to find the characteristic polynomial of f. аo аo 0 а, Char( det(A-x1 =det 0 0 3 (а, — х)(а - х) -(а, — х)

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