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Linear Algebra: Linear Systems and Matrices - Quadratic ...

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") ,westu

Linear equations are important since non-linear, di erentiable functions can be approximated by linear ones (as we have seen). orF example, the behavior of a di erentiable function f: R2!R around a point x can be approximated by the tangent plane at x. The equation for the tangent plane is one linear equation in two ariablevs.

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