Lecture
Download
1 / 22

Lecture 4 - PowerPoint PPT Presentation


  • 120 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Lecture 4' - kohana


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Lecture 4

Lecture 4

Solving simple stoichiometric equations

A linear system of equations

TheGaußscheme

Multiplicativeelements.

A non-linear system

Matrix algebra dealsessentiallywithlinearlinear systems.


Lecture 4

Solving a linear system

Thedivisionthrough a vectoror a matrixis not defined!

2 equations and fourunknowns


Lecture 4

Determinants

Det A: determinant of A

The determinant of linear dependent matricesis zero.

Suchmatricesarecalledsingular.


Lecture 4

Higher order determinants

for any i =1 to n

Laplace formula

Thematrixislinear dependent

Thenumber of operations raiseswiththefaculty of n.


Lecture 4

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.


Lecture 4

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


Lecture 4

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


Lecture 4

Systems of linearequations

Determinant


Lecture 4

Solving a simplelinear system


Lecture 4

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


Lecture 4

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


Lecture 4

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


Lecture 4

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


Lecture 4

We haveonlyfourequations but five unknowns.

The system isunderdetermined.

Themissingvalueisfound by dividingthevectorthroughitssmallestvalues to findthesmallestsolution for natural numbers.


Lecture 4

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.


Lecture 4

Thematrixissingularbecause a1, a7, and a10 do not containnewinformation

Matrix algebra helps to determinewhatinformationisneeded for an unequivocalinformation.

Fromtheknowledge of thesalts we get n1 to n5


Lecture 4

We havesixvariables and sixequationsthatare not contradictory and containdifferentinformation.

Thematrixistherefore not singular.


Lecture 4

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


Lecture 4

Population growth

We have an overshot.

In thenext time step thepopulationshoulddecreasebelowthecarryingcapacity.

Overshot

K

N

K/2

t

Fastestpopulation growth