The book of nature is written in
the language of mathematics
Galileo Galilei
Modelling Biology
Modelling Biology
Basic Applications of Mathematics and
Basic Applications of Mathematics and
Statistics in the Biological Sciences
Statistics in the Biological Sciences
Part II: Data Analysis and Statistics
Part I: Mathematics
Modelling Biology
Script A
Script A
Basic Applications of Mathematics and
Statistics in the Biological Sciences
Introductory Course for Students of
Introductory Course for Students of
Biology, Biotechnology and Environmental Protection
Biology, Biotechnology and Environmental Protection
Part I: Mathematics
Script B Werner Ulrich
Werner Ulrich
Introductory Course for Students of
Biology, Biotechnology and Environmental Protection
Werner Ulrich
UMK Torun
UMK Toruń
2007
2007
UMK Toruń
2007
www.uni.torun.pl/~ulrichw
Additional sources
http://en.wikipedia.org/wiki/Matrix_(mathematics)
K. Kaw. 2002. Introduction to matrix algebra
http://www.autarkaw.com/books/matrixalgebra/index.html
http://www.ems.bbk.ac.uk/faculty/phdStudents/efthyvoulou/Kaw.pdf
Introduction to matrix algebra and linear models:
http://nitro.biosci.arizona.edu/courses/EEB581-2006/handouts/LinearI.pdf
http://matwww.ee.tut.fi/Kost/MatrixAlgebra-toc.html
Matrix cook book
http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf
Matrix
http://en.wikipedia.org/wiki/Matrix_theory
A first course in linear algebra (free online textbook)
http://linear.ups.edu/download.html
Matrix algebra and regression
http://www.stat.tugraz.at/courses/files/s05.pdf
Mathe online
http://www.mathe-
online.at/mathint.html
Our program
1. Vectors and linear transformations
This compact lecture centres on
matrix algebra and its applications
2. Matrices in biology, data bases, basic operations
in biology.
3. Solving linear systems
For each lecture I ll give the
4. Linear and multiple regression
concepts and key phrases to get
5. Eigenvalues and eigenvectors
acquainted with.
6. Markov chains
7. Other applications
Vectors
y A vector marks a shift of a point at y
0
ć
SS = = 0
position P to position Q.
0
Ł ł
Q
S
Q
Dx
ć
v =
P
Dy
Ł ł
Dy
v
P
v v
Dx
x
x
xq - xp
ć xp - xq
P (xP;yP) ć
xq - xp
ć
PQ =
QP =
PQ =-QP
PQ =
yq - yp
yp - yq
Q (xQ;yQ) Łł
yq - yp
Łł
Łł
Vectors of the same length and direction are identical.
Vectors are either
denoted with bold type small letters: u, v, w&
or as segment lines with an arrow: PQ
Hermann Gnther Gramann
1809-1877
b
ć
u =
y i and j are called unit
a
Ł ł
vectors.
u
3
b a
cosa = ;sina =
r r
b
ć
u = = bi + aj
2
a
r cosa
ć
Ł ł
u =
r sina
Ł ł
r
1
u
a
Polar coordinates
j
a
b
i
x
1
2
3 4
The length of a vector
r2 = a2 + b2
r = u = a2 + b2
x1
ć
x2
n
N-dimensional space
v = x3
2
v =
x
i
i=1
...
xn
Ł ł
Basic operations
v
a+b+c+d=c+a+b+d=e
y
y
u
xv
ć yu
v =
yv
Ł ł
d
xu
yv
v
d
yv + yu
e
c
e
xv
xv + xu
b
b
xu
ć
u
a
u =
a
c
yu
Ł ł
x
x
Inequation of the triangle
xu xv xu + xv
ć ć ć
u + v = + =
v
yu yv yu + yv
u
Ł ł Ł ł Ł ł
u+v
u+v = v+u Commutative law
xu xv
ć ć
u =
u+o=u Zero element
; v =
yu yv
Ł ł Ł ł
u+(v+w)=(u+v)+w Associative law
u + v = (xu + xv )2 + (yu + yv )2 Ł u + v = (xu + xv )2 + (yu + yv )2
u+(-u)=o Additive inverse
Basic operations
xu u
ć
u =
a-b-c-d=a-(b+c+d)=e
y
yu
Ł ł
y
Xu + xv
yu - yv
yu e
d
u
v
d
yv
e
xu
xv
c
xv
ć
a
c
v
v =
yv
Ł ł a
b
b
x
x
xu xv xu - xv
ć ć ć
a-b-c=a-(b+c)
u - v = - =
yu yv yu - yv a-o=a Zero element
Ł ł Ł ł Ł ł
a-b`"b-a
xu - xv xu - xv
ć ć ć
u - v = u + (-v) = + =
yu yu - yv
Ł ł Ł- yv ł Ł ł
The S product
x1 nx1
ć ć
y
x2 nx2
4xu
ć
nv = n x3 = nx3 ;n R
4u =
4y
Ł u ł
... ...
4yu
xn nxn
Ł ł Ł ł
u
yu
a 1 0
ć ć ć
u = = a + b = ai + bj
4xu
xu
j b
Ł ł Ł0ł Ł1 ł
i
x
Addition, subtraction, and S-product define a
so-called linear vector space.
nu = un Commutative law
A vector space is a commutative group:
1u=1 Neutral element
1. The commutative and associative laws
0u=0
hold.
nku=knu Associative law
2. A neutral element exists
n(u+v)=nu+nv Distributive law
3. An inverse element exists.
(n+k)u=nu+ku Distributive law
The S product
xu / 2
ć
u
=
u
2 yu / 2
Ł ł
u = xu 2 + yu 2
xu 2 + yu 2 u
u
= xu 2 / 4 + yu 2 / 4 = =
u/2
2 2 2
The scalar product
xu xv uv = AC(rv cos(uv)) = AC * AB
ć ć
v
uv = = xuxv + yu yv
y
yu yv
Ł łŁ ł
rv
xu yu
= cos(u); = sin(u)
ru ru
xv yv
= cos(v); = sin(v)
rv rv
a = uv = v - u
v
u
cos(uv) = cos(v - u) = cos(v)cos(u) + sin(v)sin(u)
ru
C
a
xu xv yu yv
A
B
cos(uv) = +
ru rv ru rv
xu xv + yu yv = uv = rurv cos(uv)
x
The scalar or dot product between two vectors results in a scalar.
u1 v1
ć ć
u v
2 2
n
uv = vu Commutative law
u v
uv = = u1v1 + u2v2 + u3v3... = vi
3 3 u
i
u1=u Neutral element
i=1
uo=o Zero element ... ...
u v
(k+n)u=ku+nu Distributive law
n n
Ł łŁ ł
u (vw)`"(uv)w Associative law doesn t hold
The inequality of Cauchy-Schwarz
uv Ł rurv = u v
The scalar product of orthogonal vectors
y2
y = mx + b1
uv = u v cos(p / 2) = 0
a
1 y1
y = - x + b2
x1 b x2
y
m
rv
ru
v
ru
xn1
u
V
y2 - y1
= m1
ć
ru xv
b
x2 - x1
v'=
x
rv yv
xn1 - xn2 - (xn2 - xn1) Ł ł
xn2
= =
yn2 - yn1 yn2 - yn1 a
ć ć
ru xv yu
yn1 yn2
v'= =
xu ru yu
ć ć
ru
- (x2 - x1) 1
rv yv xu
uv = = (xuxv - yuxv ) = 0
Ł ł Ł- ł
= m2 = -
yu rv rv
y2 - y1 m1
Ł ł Ł- xu ł
The scalar product of orthogonal vectors is zero.
The square of a vector u2
2
uu = u u cos(0) = u u = u
ax=k a x
ć ć
ax = = ax + by = k
has an indefinite number of solutions. b
y
Ł łŁ ł
k - ax k a
Therefore, the division through a vector
y = = - x
b b b
is not defined.
Examples
What is the angle between the vectors {3,2} and
What is the direction of u that forms with v =
{4;5}?
(12;4} an angle of p/3?
uv = rurv cos(uv) y
uv
cos(uv) =
u v
rv
p/3
u
v
a
3 4
ć ć
25 =12 +10 = 9 + 4 16 + 25 *cos(uv)
x
Ł łŁ ł
xv 12
22 cosa a = arccosć 0.32
= cos(uv) uv = arccos(0.953) 0.308
rv 144 +16
Ł ł
533
p
+ 0.32 =1.37
3
Are the vectors {3,9} and {-12;4} perpendicular?
What z makes {6,0,12} and {-8,13,z} parallel?
6 - z
3 -12 ć ć
ć ć
uv =
uv = = 0 z =16
0 13
94 = 36 - 36 = 0
12
Ł łŁ ł
Ł łŁ-8ł
6 - z
ć ć
uv = = 0 z - 26
0 -8
For what z are {6,0,12} and -8,13,z}
1213
Ł łŁ ł
perpendicular?
6 -8 6 13
ć ć ć ć
uv = = 0 nd
0 - z
uv = = -48 + 0 +12z = 0
0 13
12
Ł łŁ-8ł
12
z
Ł łŁ ł
6 -8
ć ć
48
uv = = 0 nd
0 - z
z = = 4
1213
12
Ł łŁ ł
Linear dependencies
We have k vectors of the same dimension a1 to ak.
A linear combination is then the sum of these vectors of the form
k
u = l1a1 + l2a2 + l3a3 +...+ lkak = ai
li
i=1
7 2
ć ć
a1 b1 c1
ć ć ć 7 2 0
ć ć ć
1
ć
2 2
2 2 0
a b c
u = 3 - 4 + 22
u = 3 2 + 4 2 + 5 2 u = 3 - 4 - 5
5 2
5 2 0
3
a3 b3 c3
Ł ł
p 3
a b c p 3 0
Ł ł Ł ł
Ł 4 ł Ł 4 ł Ł 4 ł Ł ł Ł ł Ł ł
Vectors are linearly independent if
u = l1a1 + l2a2 + l 3a3 +...+ lkak = o
has only one solution of l1=l2=l3=& =lk = 0
Are the vectors {25,64,144}, {5,8,12}, Are the vectors {1,2,5}, {2,5,7}, and
and {1,1,1} linearly independent? {6,14,24} linearly independent?
1 2 6 l + 2h + 6 = 0
ć ć ć
25 5 1 25l + 5h + = 0
ć ć ć
2l
64l
l2 +h5 + = o + 5h +14 = 0
14
l64 +h8 + = o + 8h + = 0
1
5 7 24 5l + 7h + 24 = 0
144 12 1 144l +12h + = 0
Ł ł Ł ł Ł ł
Ł ł Ł ł Ł ł
2l + 5h +14 = 0
144l +12h + = 0
1 2 6
ć ć ć
- l +h = 0
39l +13h = 0
l2 +h5 + = o
13
14
ż39l - 80l = 0 l = 0 h = 0 = 0
5 7 24
80l + 4h = 0 4
Ł ł Ł ł Ł ł
1 2 6
ć ć ć
Vectors are linearly dependent if we can express one of
22 + 25 = 114
them as a linear combination of the others. 5 7 24
Ł ł Ł ł Ł ł
The vector product
w The vector or cross product combines two vectors to give a
third vector that is perpendicular to the plane defined by the two
factors .
u x v = w
v
w = u v sin(uv)
A = hu = u v sin(uv)
u
The length of the cross product vector equals the area of the
parallelogram made by the two factors.
w
a x b = -b x a Antisymmetry
|a x b| = |a||b|; if a and b are orthogonal
v
a x b = o; if "" ab = 0 or p
h
a x (b+c)= a x b + a x c
Distributive law
k(a x b) = ka x b = a x kb Associative law
u
a x a = o
a x o= o null element
a x 1= b no neutral element
The vector product
w
a d (bf - ce)
ć ć ć
uxv = =
b xe (cd - af ) = w
c (ae - bd)
f
Ł ł Ł ł Ł ł
v
a (bf - ce)
ć ć
h uw =
b (cd - af ) = abf - ace + bcd - baf + cae - cbd = 0
c (ae - bd)
Ł łŁ ł
u
a d
ć ć
u = = ai + bj+ ck; v = = di + ej + fk
b e
c
f
Ł ł Ł ł
uxv = -vxu
ixj = k;ixk = -j; jxk = i
uxv = (ai + bj+ ck)x(di + ej+ fk)=
aiej + aifk + bjdi + bjfk + ckdi + ckej = aiej + aifk - bidj + bjfk - cidk - cjek
uxv = (ae - bd)ij + (af - cd )ik + (bf - ce) jk = (ae - bd)k - (af - cd ) j + (bf - ce)i
a d (bf - ce)
ć ć ć
uxv = =
b xe (cd - af ) = w
c (ae - bd)
f
Ł ł Ł ł Ł ł
What is the volume of a tetraeder?
What is the volume of the tetraeder given
by
D
A {1,2,3}
B {2,1,4}
C {4,5,1}
h
C
D {3,4,6}
2 -1 3 2
ć ć ć
ABxA
AB = 2 AC =
C
1- ; 3 ; AD = 2
B
4 - 3 3
Ł ł Ł- 2 Ł ł
ł
A 1 3 -1
ć ć ć
ABxAC = =
-1x3 5
hF
V = 1 0
Ł ł Ł- 2 Ł ł
ł
3
-1 2
ć ć
(ABxAC) = 2F
V = / 6 = 8 / 6 = 4 / 3
5 2
(ABxAC)AD = (ABxAC) h
0 3
Ł łŁ ł
(ABxAC)AD
V =
6
Application
An aeroplane flies from Berlin to Warsaw with
constant speed of 550 kmh-1. Wind blows from
Berlin
Warsaw
the north with speed 50 kmh-1. In which
direction does the aeroplane fly? What is the
550 0
ć ć
v =
0 ;u =
new speed?
Ł ł Ł- 50
ł
550
ć
u + v = u + v = 5502 + 502 = 552
Ł- 50
ł
5502
u(u + v)= u u + v cosa cosa = = 0.99637 a = 0.085rad = 4.879o
550*552
Vectors and geometry
a -b = c (a -b)2 = a2 + b2 - 2ab = c2 a2 + b2 - 2abcos(ab) = c2
Cosine law
b
a
If a and b are orthogonal cos( "ab) = 0:
Law of Pythagoras
c
(a + b)(a -b) = a2 -b2 = a2 -b2 = 0
a-b
The vectors a+b and a-b are orthogonal.
b
a+b
a
a
Geometric projections
Reflexion about an
Parallel shift
axis
P
P
v x
a 0 a
P ć ć ć
x'= x + v = + =
b
x Ł ł Ł- 2b Ł- b
ł ł
x
v
x =x+v
x
P
Reflexion about the Stretching Turning about an angle a
origin
P
P
x =3x
u
P
v
P
P
P
a
v
- a v = u
ć
x'= -x =
uv = u v cos(a) = u2 cos(a)
Ł- b ł
uuv
= u cos(a) v = u cos(a)
u v
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