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Lecture 4. Solving simple stoichiometric equations. A linear system of equations. The Gauß scheme. Multiplicative elements . A non-linear system. Matrix algebra deals essentially with linear linear systems. Solving a linear system.

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slide1

Lecture 4

Solving simple stoichiometric equations

A linear system of equations

TheGaußscheme

Multiplicativeelements.

A non-linear system

Matrix algebra dealsessentiallywithlinearlinear systems.

slide2

Solving a linear system

Thedivisionthrough a vectoror a matrixis not defined!

2 equations and fourunknowns

slide3

Determinants

Det A: determinant of A

The determinant of linear dependent matricesis zero.

Suchmatricesarecalledsingular.

slide4

Higher order determinants

for any i =1 to n

Laplace formula

Thematrixislinear dependent

Thenumber of operations raiseswiththefaculty of n.

slide6

For a non-singularsquarematrixtheinverseisdefined as

Singularmatricesarethosewheresomerowsorcolumnscan be expressed by a linearcombination of others.

Suchcolumnsorrows do not containadditionalinformation.

Theyareredundant.

A matrixissingularifit’s determinant is zero.

r2=2r1

r3=2r1+r2

A linearcombination of vectors

Det A: determinant of A

A matrixissingularifatleast one of theparameters k is not zero.

slide7

Theaugmentedmatrix

Thetrace of a squarematrixisthe sum of its diagonal entries.

An insectspeciesatthreelocationshasthefollowingabundances per season

The diagonal entries (trace) of thedotproduct of AB’ containthetotalnumbers of insects per sitekept by predators

Thepredationrates per seasonaregiven by

slide8

Theinverse of a 2x2 matrix

Theinverse of a diagonal matrix

Determinant

Theinverse of a squarematrixonlyexistsifits determinant differsfrom zero.

Singularmatrices do not have an inverse

(A•B)-1 = B-1 •A-1 ≠ A-1 •B-1

Theinversecan be unequivocallycalculated by the Gauss-Jordan algorithm

slide11

The general solution of a linear system

Identitymatrix

OnlypossibleifAis not singular.

IfAissingularthe system has no solution.

Systems with a uniquesolution

Thenumber of independent equationsequalsthenumber of unknowns.

X: Not singular

TheaugmentedmatrixXaugis not singular and hasthe same rank as X.

Therank of a matrixis minimum number of rows/columns of thelargestnon-singularsubmatrix

slide12

A matrixislinear independent ifnone of theroworcolumnvectorscan be expressed by a linearcombinations of theremainingvectors

A linearcombination of vectors

A matrix of n-vectors (roworcolumns) iscalledlinear dependent ifitispossible to express one of thevectors by a linearcombination of theother n-1 vectors.

r2=2r1

r3=2r1+r2

Thematricesarelinear dependent

If a vectorV of a matrixislinear dependent on theothervecors, Vdoes not containadditionalinformation. Itiscompletelydefined by theothervectors. ThevectorVisredundant.

Linearindependence

slide13

How to detectlineardependency

Any solution of k3=0 and k1=-2k2satisfiestheaboveequations. Thematrixislinear dependent.

If a matrixAislinearly independent, thenanysubmatrix of Aisalsolinearly independent

Therank of a matrixisthemaximumnumber of linearly independent row and columnvectors

slide15

Consistent

Rank(A) = rank(A:B) = n

Infinitenumber of solutions

Consistent

Rank(A) = rank(A:B) < n

No solution

Inconsistent

Rank(A) < rank(A:B)

Infinitenumber of solutions

Consistent

Rank(A) = rank(A:B) < n

Inconsistent

Rank(A) < rank(A:B)

No solution

Consistent

Rank(A) = rank(A:B) = n

Infinitenumber of solutions

slide17

We haveonlyfourequations but five unknowns.

The system isunderdetermined.

Themissingvalueisfound by dividingthevectorthroughitssmallestvalues to findthesmallestsolution for natural numbers.

slide18

Includinginformation on thevalences of elements

Equality of atomsinvolved

We have 16 unknows but withoutexperminetnalinformationonly 11 equations.

Such a system isunderdefined.

A system with n unknownsneedsatleast n independent and non-contradictoryequationsfor a uniquesolution.

If ni and aiareunknowns we have a non-linearsituation.

We eitherdetermine ni oraiormixedvariablessuchthat no multiplicationsoccur.

slide19

Thematrixissingularbecause a1, a7, and a10 do not containnewinformation

Matrix algebra helps to determinewhatinformationisneeded for an unequivocalinformation.

Fromtheknowledge of thesalts we get n1 to n5

slide20

We havesixvariables and sixequationsthatare not contradictory and containdifferentinformation.

Thematrixistherefore not singular.

slide21

Linearmodelsinbiology

The logistic model of population growth

K denotesthemaximumpossibledensity under resourcelimitation, thecarryingcapacity.

rdenotestheintrinsicpopulation growth rate. Ifr > 1 thepopulationgrowths, atr < 1 thepopulationshrinks.

t N

1 1

2 5

3 15

4 45

We needfourmeasurements

slide22

Population growth

We have an overshot.

In thenext time step thepopulationshoulddecreasebelowthecarryingcapacity.

Overshot

K

N

K/2

t

Fastestpopulation growth

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