Matemáticas II: Espacio Euclidiano n-dimensional y normas, Ejercicios de Matemática Financiera

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Grau en Economia

Grau en Administraci´o i Direcci´o

d’Empreses

Mathematics II

Problem set 1: The n-dimensional Euclidian space

Departament d’Economia i d’Historia Economica

PS 1

Mathematics II

J. Gierlinger

1. Determine the norm of the following vectors:

a)

v = (1, 2) b)

v = (3, 4) c)

v = (− 1 , 2)

d)

v = (− 1 , 0 , 3) e)

v = (− 2 , − 1 , −3) f)

v = (0, 0 , −3)

g)

v = (1, 2 , − 1 , 4) h)

v = (3, 0 , − 2 , 8) i)

v = (− 1 , 6 , − 2 , 4)

2. Determine the distance between the following pairs of points:

a) P = 3, Q = 1 b) P = (1, 2), Q = (1, −2)

c) P = (1, 2), Q = (− 2 , 3) d) P = (0, 1), Q = (− 1 , −2)

e) P = (3, 2), Q = (− 3 , −2) f) P = (1, 1), Q = (6, 3)

g) P = (1, 0 , 3), Q = (− 1 , − 1 , 2) h) P = (1, 1 , −1), Q = (1, 0 , 3)

i) P = (1, − 1 , 0 , 6), Q = (1, 2 , 1 , 0)

3. Calculate the scalar product of the following pairs of vectors:

a)

v = (1, 2),

w = (3, 4) b)

v = (0, 1),

w = (1, −1)

c)

v = (1, 1),

w = (1, −1) d)

v = (2, 1),

w = (4, 2)

e)

v = (3, −1),

w = (− 1 , 2) f)

v = (2, 0),

w = (0, 1)

g)

v = (1, 0 , 2),

w = (1, 2 , 1) h)

v = (− 1 , 1 , 1),

w = (2, 0 , 2)

i)

v = (1, 7 , −2),

w = (0, − 2 , −1)

4. Alex buys groceries. 3 kg of cherries, 4 kg of pears, 3 kg of oranges and 5

kg of apples. Beth buys (in kg) 2 of cherries, 2 of oranges and 4 of apples.

The price per kg is: pears 0.60e, cherries 0.75e, apples 0.40e and oranges

0.50e.

6. A set C ⊆ R

n is convex if for all sets of points x, y ∈ C, the connecting

segment xy := {z ∈ R

n |z = λx + (1 − λ)y, 0 ≤ λ ≤ 1 } is included in

set C. Check if the set

A = {(x, y, z) ∈ R

3 | x + y + z = 1}

is convex.

7. Recall that a function is convex if and only if the set of all points above the

graph is a convex set. Similarly, a function is concave if and only if the set

of all points below its graph is a convex set. Use these results to see whether

the following sets are convex.

a) A = {(x, y) ∈ R

2 | x

2

• y

2 ≤ 1 }.

b) A = {(x, y) ∈ R

2 | x ≥ y

2 }.

c) A = {(x, y) ∈ R

2 | y ≥ e

2 x }.

d) A = {(x, y) ∈ R

2 | x − y ≤ 2 }.

e) A = {(x, y) ∈ R

2 | y ≥ x

4

• 2x

2 − x − 3 }.

f) A = {(x, y) ∈ R

2 | y ≤ sin(x)}.

g) A = {(x, y) ∈ R

2 | x ≥ y

3 }.

h) A = {(x, y) ∈ R

2 | y ≥

1

x^2

i) A = {(x, y) ∈ R

2 | (y

2

• 1)x ≥ 1 , y ≥ x

2 }.

j) A = {(x, y) ∈ R

2 | x ≤ −y

6 − 3 x

2

• 12, y ≥ 4 }.

8. Let A and B be convex sets in R

n

. Define the set A + B as (note that this is

not the union):

A + B = {z ∈ R

n | z = x + y, x ∈ A, y ∈ B}

Show that the new set A + B is also convex.