MATH 2: ANSWERS WEEK 6 (optional parts)
LINEAR ALGEBRA
p = 1/2
1:
2: a. (
)
b.
( )(
) =0
(
)(
) =0
3: a. (
)
b.
c, ( ) + (
4 a. (
(
)=(
)
)
)
b. 2x y = 3
5: a. (
b. x + y = 1
)
6. 2x + y + 3z = 13
LANGUAGES
a,b
6.1:
cfw_
b
a
cfw_S0
b
a
b
cfw_S1
a
cfw_S1,S

MATH 2 week 3
READER LINEAR ALGEBRA
Chapter 3:
- Introduction to vectors
- Vector form of a line.
MATH 2 week 3: vectors
A vector is a matrix with only one column:
or
A vector can be depicted as an arrow in 2D or 3D.
Y-axis
Such an arrow has:
- a length

Languages, grammars and
automata.
week 2
This week: Chapter 12
- 12.4 Regular expressions, regular languages.
.
languages
There are different ways to describe a language:
A set notation:
L1 = cfw_ abn | n >=0 or L2 = cfw_a, b* cfw_ab
a graph (next week

Languages, grammars and
automata
week 4
This week:
- Finite state automaton (section 12.5 till pumping lemma)
- From graph to state diagram of the finite state automaton
Repetition: Regular languages
Give a graph for the language described by:
*
(0 1) 00

Languages, grammars and
automata.
week 6
converting a graph into a finite state automaton
a non-deterministic graph for (ab)*a (ab)*b
can you find a path such that abab is not accepted?
this week: how to make a FSA from this graph?
a,b
s0
a,b
a
s1
b
s2
2

FONTYS
INFORMATION & COMMUNICATION TECHNOLOGY
ENGLISH COURSE
Course
: MATH 2
Teacher
:
Date
: 16-01-2015
Time
: 08:45 10:25
Accepted resources: None
exercise
points:
1
20
2
16
3
17
4
15
5
20
6
12
If you received a confirmation from your MATH-teacher that

EXERCISE 1
a. intersection point exists due to the fact that rc ( l ) different from rc ( m ), intersection point (1, 3)
b. for example: ( x ) = ( 1 ) + lambda ( 1 )
(y) (3)
(4)
c. ( 1 ) . ( 4 ) = 4 6 = -2 = sqrt( 1 + 4 + 4) * sqrt( 16 + 9 ) * cosine( alf

1.
a. lambda=-2, mu=3, so (x,y)=(-1,13)
b. 3*-1+-4*2=5*sqrt(5)*cos(a); so cos(a)=-11/(5*sqrt(5)
2.
a. AB = 0 8
1 3
b. AC = -10 24 -2
2 -5 1
3.
a. 13x + 10y - z = 67
b. x
-2
13
y = -3 + * 10
z
8
-1
4.
a.
1 5 7
3 4 5
(
b. AX =
A-1AX
IX
X
X =
)
B
= A-1B
= A-

Languages, grammars and
automata
week 5
Today we look at
Finite state machines, section 13.2
Repetition: Regular languages
Give a graph for the language described by:
*
(0 1) 00
1
Give a regular expression
for the language given by
0
s0
1
s1
0
1
s2
0
Fin

Extra exercises typing constraints
PRESIDENT = (
pres_name
CHAR(20),
birth_yr
NUMBER(4),
yrs_serv
NUMBER(2),
death_age
NUMBER(2),
party
CHAR(10),
state_born
CHAR(15),
primary key (pres_name),
);
presidents
name of the president
birth year
years of service

EDB2 Week Overview
Datamodeling and Design
Book: Database Processing by David Kroenke and David Auer (11 th edition or 12th edition)
Schema 12th edition: (11th edition on the next page!)
Week 1
Study:
Book: chapter 5 till page 196
Exercises
Review questio

MATH 2 week 4
READER LINEAR ALGEBRA
Chapter 4:
- Planes
- Vector form of a plane
MATH 2 week 4: usage of vectors
MATH 2 week 4: repetition week 2
A line in 2D can be represented by
- a linear equation, for example: 3x + 4y = 12
or
- a vector-form, for exa

Languages, grammars and
automata
week 3
This week: Chapter 12.4
- Graph of a regular languages
Repetition: Regular languages
Exercise:
r1=aa*b bb*a
r2=(aaa)*b
Give 3 words of each of the languages L(r1) and L(r2)
Give the set notation of both languages.

MATH 2: ANSWERS WEEK 5 (optional parts)
LINEAR ALGEBRA
5.1:
8
5.2: a. (
b.
)
(
)
5.3: a. ( ) = (
(
)
)+( )
b. dot prducts equals 0
5.4 a. ( ) = ( ) + ( )
b. (27/13, 18/13)
5.5: a .
b. (6/
4/
, 8/
LANGUAGES
5.1: a
S0
S1
S2
b. (i) xzzxxyyx
a
S1x
S0,z
S2,y

MATH2 WEEK 6: EXERCISES
LINEAR ALGEBRA
Exercise 1
For which p the vectors (
) and ( ) are perpendicular?
Exercise 2
Let a =(
) and b = ( ).
a. Compute a b.
b. Check with the dot product that a b is perpendicular to a and to b
Exercise 3
a = ( ) and b = (

MATH2 WEEK 4: EXERCISES
LINEAR ALGEBRA
Exercise 4.1
a=(
)
b=(
)
For which p and q are a and b dependent?
Exercise 4.2
Given :
( )
=
( )+ ( )
+ (
)
Why is this not a vector-form of a plane?
Exercise 4.3
Plane V2 goes through (2, 3, 6) and line l: (
Give a

MATH2 WEEK 3: EXERCISES
LINEAR ALGEBRA
EXERCISE 3.1
Let A = (3,1,5) and B = (2,0,9). Give vector AB.
EXERCISE 3.2
l is the line with equation y = 3x + 5.
a. Give a vector that is parallel to line l.
b. Give a vector-form of the line.
EXERCISE 3.3
a = ( )

MATH2 WEEK 1: EXERCISES
LINEAR ALGEBRA
Exercise 1.1
Which of the following equations are linear?
a. y = x/2 + 3 a.
c. y = x2 + 3
b. y = 2/x + 3
d. y = 5 + 3x
c.
b.
d.
Exercise 1.2
Solve the following pair of simultaneous equations by elimination:
7x+y=25

MATH2 WEEK 5: EXERCISES
LINEAR ALGEBRA
Exercise 5.1
Given two vectors (
) and ( ).
Calculate the dot product of these vectors.
Exercise 5.2
a. Find a vector that is perpendicular to (
).
b. Find two independent vectors that are both perpendicular to (
Exe

MATH2 WEEK 2: EXERCISES
LINEAR ALGEBRA
Exercise 2.1
Consider the next matrices:
A=
B=
D=(
)
C =
E=(
)
Calculate if possible:
a. D-E
b. CA
c. AC
d. D -1
e. 2E 3D
Exercise 2.2
Which of the five given matrices in the previous exercise are unity matrices?
Exe

MATH 2: ANSWERS WEEK3 (optional parts)
LINEAR ALGEBRA
(
3.1:
3.2:a. (
)
) or ( )
b. ( )
(
)
3.3: a.
b. y =
3.4: No
3.5. y =
3.6: (
LANGUAGES
b
3.1: a.
a
S1
b
S2
a
S3
S4
3.1: b.
a
a
S2
S1
a
3.1: c.
a*ba*
S3
b
S1
3.2:
b
b
b
S2
or cfw_a*cfw_bcfw_a*
3.3: b*(a

MATH 2: ANSWERS WEEK1 (optional parts)
LINEAR ALGEBRA
1.1: a. is linear (the equation can be written as: y = x +3) and d. is linear.
b. and c. are not linear.
1.2: 7x + y = 25
5x y = 11
+
12 x = 36
x =3
21 + y = 25 y = 4
So, the solution is x=3, y=4
1.3

MATH 2: ANSWERS WEEK2 (optional parts)
INEAR ALGEBRA
2.1: a.
(
)
d. (
)
b. not possible
e. (
c. (
)
2.2: Only E
2.3: a. (
)
b. 19
c. one
d. (1/19, 31/19)
2.4: a. (
)
b. 0
c. zero
d. there are no solutions
2.5: a. (
)
b. 0
c. more
d. the whole line: 2x 3y=

MATH2 (201504)
lecturers:
Paul Linnartz
Ivan Zapreev
Marco Dorenbos
Materials for Math2
Book: Discrete mathematics
(seymout lipschutz/marc lipson)
The reader: LINEAR ALGEBRA
available on Sharepoint
Also on Sharepoint:
course description, exercises, pla