MT 27 - Autumn 2016
Final Exam - January, the 16th 2016
Exercise 1What is the period and the symmetries of the parametric curve defined byx(t) = cost 1 + sin2t and y(t) = sintcost
1 + sin2t.
Exercise 2Compute det
2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2
by applying row operations to produce an upper triangular
matrix.
Exercise 3 Prove that if two diagonalizable matrices A and B share the same eigenvector matrix then AB=BA.
Exercise 4 A door is opened between two rooms that holdv(0) = 30 people and w(0) = 10 people.
The movement between rooms is assumed to be proportional to the difference v−w:
dv
dt =w−v and dw
dt =v−w.
a. Without computing the solution, show that the total v+w of people in the two rooms is constant (40 people).
b. Write the system under the form of a system of differential equation du
dt =Auwith matrixA, solve this system and find v and watt= 1.
Exercise 5We consider the parametric curvex(t) = t3
t2−9 andy(t) = t(t−2)
t−3 .Find the asymptotes att=±∞.
Exercise 6 Compute the minimum min
(a,b)∈R2
Z 1 0
(t2−at−b)2 dt.
Exercise 7 Use the ratio test to test for convergence or divergence of the series a. P n!
3n and b.
P n 2n.
Exercise 8 Give fast answers to the following questions.
a. If a 4×4 matrix has detA= 12, find det(2A), det(−A), det(A2) and det(A−1).
b. If the eigenvalues ofA are 1 and 0, write everything you know about the matrices Aand A2.
Exercise 9 For each statement, shortly explain why it is true or false.
a. If Aand B are identical except their first entryb11= 2a11, then detB = 2 detA.
b. The determinant ofAB−BAis zero.
c. It is possible that a square matrix A is diagonalizable and that there exists a constant c so that A+cI is not (I being the identity matrix).
d. IfP
an converges thenP|an|converges.
e. IfP
an converges then limn→∞|an+1|/|an| 6= 1.
f. The seriesP
(−1)n2n converges.
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Exercise 10*In some cases, an invertiblen×nmatrixA∈Rn×n and a vectorb∈Rn are such that the solutionx∈Rn of the linear system
Ax=b
is not stable under small perturbations ofb. For instance forA=
10 7 8 7
7 5 6 5
8 6 10 9
7 5 9 10
and b=
32 23 33 31
the solution is x =
1 1 1 1
. If we perturb slightly the right-hand side,b0 =
32.1 22.9 33.1 30.9
the new solution
turns out to bex0 =
9.2
−12.6 4.5
−1.1
. In other words, a relative error of the order 0.1/20 = 0.005 in the data
b produces a relative error of the order 10/1 = 10 in the solution, which represents an amplification of the relative error of the order 10/0.005 = 2000. This exercise explains why.
a.Posing b0 =b+δband x0=x+δx prove thatAδx=δb.
Let||x||= q
P4
i=1x2i be the euclidean norm of any vector x= (x1, x2, x3, x4)T ∈R4, we admit that the mapping defined from the set M4(R) of the 4×4 matrices to R,
A7→ sup
x∈R4
||Ax||
||x||
is a norm denoted ||A|| = supx∈R4||Ax||/||x||. We also recall that the euclidean norm is associated to the inner product hx, yi=P4
i=1xiyi. Thus ||A||= supx∈R4
phAx, Axi/||x||. In the following we assume that the matrix A is symmetric.
b*.Prove that for any 4×4 symmetric matrix Athe inequality||A||2 ≤λ2maxholds whereλmaxstands for the largest eigenvalue of A. With a special choice of x in the definition of the norm, deduce that
||A||=|λmax|.
c*. Prove that ||A−1||= 1
|λmin| where λmin stands for smallest eigenvalue ofA.
d*.Prove that ||Ax|| ≤ ||A|| ||x||for any A and x, and deduce that
||δx||
||x|| ≤ ||A−1|| ||A||||δb||
||b||
e. Explain the instability result provided that the eigenvalues of the matrixAgiven in the introductory example are λ1 ≈ 30, λ2 ≈ 3.8, λ3 ≈0.84 and λ4 ≈0.01. Finally, the ratio λmax/λmin being called the condition number of the matrix A, state a condition that garanties a small instability in the resolution of a linear system.
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