The book of nature is written in the language of mathematics

1. The book of nature is written in the language of mathematics Galileo Galilei

2. www.uni.torun.pl/~ulrichw

3. Additionalsources 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 linearmodels: 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 courseinlinear algebra (freeonlinetextbook) http://linear.ups.edu/download.html Matrix algebra and regression http://www.stat.tugraz.at/courses/files/s05.pdf

4. Matheonlinehttp://www.mathe-online.at/mathint.html

5. Our program 1. Vectors and lineartransformations 2. Matricesinbiology, data bases, basic operations 3. Solvinglinear systems 4. Linear and multipleregression 5. Eigenvalues and eigenvectors 6. Markovchains 7. Otherapplications This compact lecture centres on matrix algebra and itsapplicationsinbiology. For each lecture I’ll give the concepts and key phrases to get acquainted with.

6. Vectors A vector marks a shift of a point at position P to position Q. y y Q S Q P Dy v P v v Dx x x P (xP;yP) Q (xQ;yQ) 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 GüntherGraßmann 1809-1877

7. i and jarecalled unit vectors. y u 3 2 r 1 u a Polar coordinates j a b i x 1 2 4 3 Thelength of a vector N-dimensionalspace

8. Basic operations v a+b+c+d=c+a+b+d=e y y u yu d xu yv v d yv + yu e c e xv xv + xu b b u a a c x x Inequation of the triangle v u u+v u+v = v+u Commutative law u+o=u Zero element u+(v+w)=(u+v)+w Associative law u+(-u)=o Additiveinverse

9. Basic operations u a-b-c-d=a-(b+c+d)=e y y Xu+ xv yu- yv yu e d u v yv d e xu xv c a c v a b b x x a-b-c=a-(b+c) a-o=aZero element a-b≠b-a

10. The S product y 4yu u yu 4xu xu j i x Addition, subtraction, and S-productdefine a so-calledlinearvectorspace. A vectorspaceis a commutative group: Thecommutative and associativelawshold. A neutral element exists An inverse element exists. nu = unCommutative law 1u=1 Neutral element 0u=0 nku=knuAssociative law n(u+v)=nu+nvDistributive law (n+k)u=nu+kuDistributive law

11. The S product u u/2

12. Thescalarproduct v y rv v u ru C a A B x Thescalarordotproductbetweentwovectorsresultsin a scalar. uv = vu Commutative law u1=uNeutral element uo=o Zero element (k+n)u=ku+nuDistributive law u (vw)≠(uv)w Associative lawdoesn’thold Theinequality of Cauchy-Schwarz

13. Thescalarproduct of orthogonalvectors y2 a y1 x1 b x2 y rv ru v ru xn1 u V’ b x xn2 a yn1 yn2 Thescalarproduct of orthogonalvectorsis zero. Thesquare of a vectoru2 ax=k has an indefinitenumber of solutions. Therefore, thedivisionthrough a vectoris not defined.

14. Examples Whatistheanglebetweenthevectors {3,2} and {4;5}? Whatisthedirection of uthatformswithv = (12;4} an angle of p/3? y rv p/3 u v a x Arethevectors {3,9} and {-12;4} perpendicular? What z makes {6,0,12} and {-8,13,z} parallel? For what z are {6,0,12} and -8,13,z} perpendicular?

15. Lineardependencies We have k vectors of the same dimensiona1 to ak. A linearcombinationisthenthe sum of thesevectors of the form Vectorsarelinearly independent if hasonly one solution of l1=l2=l3=…=lk = 0 Arethevectors {1,2,5}, {2,5,7}, and {6,14,24} linearly independent? Arethevectors {25,64,144}, {5,8,12}, and {1,1,1} linearly independent? Vectorsarelinearly dependent if we canexpress one of them as a linearcombination of theothers.

16. The vector product The vector or cross product combines two vectors to give a third vector that is perpendicular to the plane defined by the two factors . w u x v = w v 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 a x b = o; if ∢ ab = 0 or p a x (b+c)= a x b + a x c Distributive law k(a x b) = ka x b = a x kb Associative law a x a = o a x o= o null element a x 1= b no neutral element v h u

17. Thevectorproduct w v h u

18. Whatisthevolume of a tetraeder? Whatisthevolume of thetetraedergiven by A {1,2,3} B {2,1,4} C {4,5,1} D {3,4,6} D h C ABxAC B A

19. Application An aeroplane flies from Berlin to Warsaw with constant speed of 550 kmh-1. Wind blows from the north with speed 50 kmh-1. In which direction does the aeroplane fly? What is the new speed? Berlin Warsaw

20. Vectors and geometry Cosine law b a Ifa and bareorthogonal cos(∟ab) = 0: Law of Pythagoras c a-b Thevectorsa+b and a-bareorthogonal. b a+b a a

21. Geometricprojections Reflexionabout an axis Parallelshift P P’ v x P x’ x v x’=x+v x’ P’ Reflexionabouttheorigin Stretching Turningabout an anglea P’ P’ x’=3x u P v P P’ P a v