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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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Lecture 4

Solving simple stoichiometric equations

A linear system of equations

TheGaußscheme

Multiplicativeelements.

A non-linear system

Matrix algebra dealsessentiallywithlinearlinear systems.


Solving a linear system

Thedivisionthrough a vectoror a matrixis not defined!

2 equations and fourunknowns


Determinants

Det A: determinant of A

The determinant of linear dependent matricesis zero.

Suchmatricesarecalledsingular.


Higher order determinants

for any i =1 to n

Laplace formula

Thematrixislinear dependent

Thenumber of operations raiseswiththefaculty of n.


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.


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


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


Systems of linearequations

Determinant


Solving a simplelinear system


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


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


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


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


We haveonlyfourequations but five unknowns.

The system isunderdetermined.

Themissingvalueisfound by dividingthevectorthroughitssmallestvalues to findthesmallestsolution for natural numbers.


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.


Thematrixissingularbecause a1, a7, and a10 do not containnewinformation

Matrix algebra helps to determinewhatinformationisneeded for an unequivocalinformation.

Fromtheknowledge of thesalts we get n1 to n5


We havesixvariables and sixequationsthatare not contradictory and containdifferentinformation.

Thematrixistherefore not singular.


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


Population growth

We have an overshot.

In thenext time step thepopulationshoulddecreasebelowthecarryingcapacity.

Overshot

K

N

K/2

t

Fastestpopulation growth


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