Hello, I would like help on a couple linear algebra questions, please. Just part 2 of this homework, I figured out part 1. Thank you!

 

 

#3, Due Monday 08/22 at 6pm
Last updated on 08/15/2016 Part 1
Section 4.3: 10, 11, 12
Section 5.1: 1(a)(b)(d)(e)(f)(i)(j)(k), 3(a)(b)(c),4(f),8(a)(b),17(a)(b)(c)
Section 5.2: 1(a)-(g)(All of them!), 2(a),3(a),20
Section 5.4: 1(a)(f)(Hint: See Section 4.3, Exercise 24), 2(a)(c)(e), 18(a)(b)(c)
Section 7.1: 3(a)(c) (You only need to find the Jordan canonical form of T in (a) and
(c))
Section 7.2: 1(a)(d)(g), 3(a)-(e)(All of them!), 3(a)(c)(d)(e) Part 2
1. Prove that for any A, B ∈ Mn×n (F ), AB and BA have the same eigenvalues. More
precisely, for any λ ∈ F , λ is an eigenvalue of AB if and only if λ is an eigenvalue of BA.
2. T : V → V is a linear transformation, x ∈ V such that T k−1 (x) 6= 0, T k (x) = 0.
Prove that β = {x, T x, T 2 (x), · · · , T k−1 (x)} is linearly independent. Let W = Spanβ,
then W is a T -invariant subspace, what is [T |W ]β ?
3. Find an explicit formula for an where an is defined by
a1 = a2 = 1 and
an = an−1 + an−2
for n ≥ 3. 1

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• 1492660708-Homework 3.pdf