Transcription of Linear Algebra: Linear Systems and Matrices - Quadratic ...
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LinearAlgebra:LinearSystemsandMatrices-Q uadraticFormsandDe niteness-EigenvaluesandMarkovChainsJoshu aWilde,revisedbyIsab elTecu,TakeshiSuzukiandMar aJos Bo ccardiAugust13, +a12x2+ +a1nxnb2=a21x1+a22x2+ + +am2x2+ +amnxnLinearequationsareimp ortantsincenon- Linear ,di erentiablefunctionscanb eapproximatedbylinearones(aswehaveseen). Forexample,theb ehaviorofadi erentiablefunctionf:R2 Raroundap ointx canb eapproximatedbythetangentplaneatx . , ethoughtofasapproximationsformorecomplic atedunderlyingrelationshipsb ewritteninmatrixform: m 1= amn m n n 1,Inshort,wecanwritethissystemasb=Axwher eAisanm nmatrix,bisanm 1vectorandxisann ,alsoreferredtoaslinearmap,canthereforeb eidenti edwithamatrix,andanymatrixcanb eidenti edwith("turnedinto") ,westudymatricesandtheirprop erationsandProp ertiesConsidertwon mmatrices:A= anm , B= bnm 12 LinearAlgebraThenthebasicmatrixop +B= a11+ a1m+ + anm+bnm 2.
x is an n 1 vector. A system of linear equations , also referred to as linear map, can therefore be identi ed with a matrix, and any matrix can be identi ed with ("turned into") a linear system. In order to study linear systems, we study matrices and their properties. 2 Matrices 2.1 Basic Matrix Operations and Properties Consider two n ...
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