Tentamen i Matematik 3: M0031M.
Datum: 2009-10-26 Skrivtid: 09:00–14:00
Antal uppgifter: 6 ( 30 po¨ang ).
Jourhavande l¨arare: Norbert Euler Telefon: 0920-492878
Till˚atna hj¨alpmedel: Inga
Till alla uppgifterna skall fullst¨andiga l¨osningar l¨amnas.
Resonemang och utr¨akningar ska vara tydligt presenterade.
Aven endast delvis l¨¨ osta problem kan ge po¨ang.
Enbart svar ger 0 po¨ang.
ENGLISH VERSION:
Problem 1: Find all solutions of the equation z6 + 1 = 0,
where z are complex numbers and display your solutions on a circle in the complex plane.
[5 points]
Problem 2: Let T denote a transformation such that T : P2 → P4,
where P2 is the vector space of 2nd-degree polynomial functions and P4 the vector space of 4th-degree polynomial functions, such that
T : p(t) 7→ t2p(t), i) Show that T is a linear transformation.
ii) Find the matrix representation for T realtive to the bases B and C, where B = {1, 2t, −2 + 4t2} for P2
C= {1, t, t2, t3, t4} for P4
iii) Is T : p(t) 7→ p(t) + p(t)2, with p(t) ∈ P2, a linear transformation? Explain.
[5 points]
Problem 3: Consider the following vectors in R4:
y= (3, −1, 1, 13), v1 = (1, −2, −1, 2), v2 = (−4, 1, 0, 3).
i) Find the shortest distance from y to the subspace W of R4 spanned by v1 and v2.
ii) Find the orthogonal projection of y onto the orthogonal complement, W⊥, of W.
[5 points]
Problem 4: Find an orthonormal basis for the column space of the matrix A and give the rank of this matrix:
A=
3 −5 1
1 1 1
−1 5 −2
3 −7 8
.
[5 points]
Problem 5:
a) Consider the following differential equation:
x2dy
dx = y2+ 2xy .
i) Find the general solution of the given differential equation for all x ∈ ℜ using the substitution y(x) = x v(x).
ii) Solve the initial-value problem for the given differential equation, with y(1) = 2.
b) Consider the differential equation dy
dx = x+ 3y − 4 3x − y − 2. Assume the following change of variables,
z = x − 1 v(z) = y − 1,
where z is the new independent variable and v the new dependent variable. Show that the differential equation in terms of the new variables v and z is a separable 1st-order differential equation.
[5 points]
Problem 6:
Solve only one of the following three problems:
1) Show that
eix = cos x + i sin x, for all x ∈ ℜ
by considering the function
f(x) = (cos x + i sin x)e−ix and its derivative. Here i :=√
−1.
2) Let A be an n × n matrix. Prove that A is invertible if and only if λ = 0 is not an Eigenvalue for A.
3) Consider a general n-dimensional vector space V with basis B and a general m-dimensional vector space W with basis C. Let T be a linear transformation,
T := V → W,
Derive the matrix representation of T relative to B and C.
[5 points]
SWEDISH VERSION:
Problem 1:
Best¨am alla l¨osningar till ekvationen
z6+ 1 = 0
d¨ar z ¨ar komplexa tal, och markera dina l¨osningar p˚a en cirkel i det komplexa talplanet. fullst¨andigt.
[5 po¨ang]
Problem 2: L˚at T beteckna en avbildning, s˚a att T : P2 → P4,
d¨ar P2 ¨ar vektorrummet med polynom av grad 2 och P4 ¨ar vektorrummet med polynom av grad 4, s˚a att
T : p(t) 7→ t2p(t), i) Visa att T ¨ar en linj¨ar avbildning.
ii) Best¨am matrisrepresentationen av T relativt baserna B och C, d¨ar B = {1, 2t, −2 + 4t2} f¨or P2
C= {1, t, t2, t3, t4} f¨or P4
iii) ¨Ar T : p(t) 7→ p(t) + p(t)2, d˚a p(t) ∈ P2, en linj¨ar avbildning? F¨orklara.
[5 po¨ang]
Problem 3: Givet f¨oljande vektorer i R4:
y= (3, −1, 1, 13), v1 = (1, −2, −1, 2), v2 = (−4, 1, 0, 3).
i) Best¨am det kortaste avst˚andet fr˚an y till underrummet W till R4 som sp¨anns upp av v1 och v2.
ii) Best¨am den ortogonala projektionen av y p˚a det ortogonala komplementet, W⊥, till W .
[5 po¨ang]
Problem 4: Best¨am en ortonormerad bas f¨or kolonnrummet till matrisen A and ange matrisens rang:
A=
3 −5 1
1 1 1
−1 5 −2
3 −7 8
.
[5 po¨ang]
Problem 5:
a) Givet f¨oljande differentialekvation:
x2dy
dx = y2+ 2xy .
i) Best¨am den allm¨anna l¨osningen till differentialekvationen f¨or alla x ∈ ℜ genom att anv¨anda substitutionen y(x) = x v(x).
ii) L¨os initialv¨ardesproblemet f¨or den givna differentialekvationen d˚a y(1) = 2.
b) Givet differentialekvationen dy
dx = x+ 3y − 4 3x − y − 2. Anv¨and f¨oljande variabelbyten,
z = x − 1 v(z) = y − 1,
d¨ar z ¨ar den nya oberoende variabeln och v ¨ar den nya beroende varabeln. Visa att differentialekvationen uttryckt i de nya variablerna v och z ¨ar en separabel f¨orsta ordningens differentialekvation.
[5 po¨ang]
Problem 6:
L¨os endast ett av de f¨oljande tre problemen:
1) Visa att
eix = cos x + i sin x, for all x ∈ ℜ genom att betrakta funktionen
f(x) = (cos x + i sin x)e−ix och dess derivata. H¨ar ¨ar i :=√
−1.
2) L˚at A vara en n ×n matris. Visa att A ¨ar inverterbar om och endast om λ = 0 inte ¨ar ett egenv¨arde till A.
3) Givet ett allm¨ant n-dimensionellt vektorrum V med bas B och ett allm¨ant m-dimensionellt vektorrum W med bas C. L˚at T vara en linj¨ar avbildning
T := V → W,
H¨arled matrisrepresentationen av T relativt B och C.
[5 po¨ang]
