MT 27 - Autumn 2017
Final Exam - January, the 17th 2018
Exercise 1 We consider the curve parametrized byx(t) = t3
t2−9 andy(t) = t2−2t
t−3 fort∈R− {3,−3}.
a. Find the classification of the pointM(t) at t= 0.
b. Draw the behavior of the curve in a vicinity of M(0).
Exercise 2 Study the convergence of the three series a. P 1
10n b. P
ln(n+ 2
n+ 1) c. P 3n 4n−2n.
Exercise 3 LetV be a finite-dimensional vector space over R, an invertible endomorphism T ∈ L(V, V) and λ∈F\{0}. Prove that λis an eigenvalue forT if and only if λ−1 is an eigenvalue for T−1.
Exercise 4 Prove the two propositions of Lecture 2.
a. If two matricesAandB are similar with a similarity transformation matrixP such thatA=P BP−1, then the eigenvalues of AandB are the same. Moreover, their eigenvectors are related i.e. AX =λX iff BY =λY withX=P Y.
b. Suppose that A is similar toB, say there exists an invertible matrixP such thatP−1AP =B, then for every polynomial q(x), q(B) = P−1q(A)P. It follows that the eigenvalues of q(A) and q(B) are the same.
Exercise 5 Suppose that A is diagonalizable overR and Ahas only 2 and 4 as eigenvalues. Show that A2−6A+ 8I = 0.
Exercise 6 Let (e1, e2, e3) be the canonical basis of R3, and define f1 =e1+e2+e3,f2 =e2+e3 and f3 =e3.
a. Apply the Gram-Schmidt process to the basis (f1, f2, f3).
b. What do you obtain if you instead applied the Gram-Schmidt process to the basis (f3, f2, f1)?
Exercise 7 Let P ⊂ R3 be the plane containing 0 perpendicular to the vector (1,1,1). Using the standard norm, calculate the distance of the point (1,2,3) toP. Explain in detail all the steps.
Exercise 8 Letf the 2π-periodic function defined by
∀x∈]−π, π], f(x) =|x|.
a. Calculate the Fourier coefficients off. b. Deduce the sum P 1
(2p+ 1)4.
c. Justify the convergence of the seriesP 1
n4 and find its sum.
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