a Show that the three functions xx e are independent vectors in Hint take the derivative of a linear combination of these functions b Show that the two functions sin are independent vectors in 2 Let denote a 64257eld Recall that denotes the vec ID: 75678
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MAT310:HW3(1)RecallthatC()denotesthevectorsspaceofallcontinuousrealvaluedfunctionsde nedonalloftherealline.(a)Showthatthethreefunctionsx;x2;exareindependentvectorsinC().(Hint:takethederivativeofalinearcombinationofthesefunctions.)(b)Showthatthetwofunctionsex;sin(x)areindependentvectorsinC().(2)LetFdenotea eld.RecallthatP2(F)denotesthevectorspaceofpolynomialsinonevariablehavingdegreelessthanorequalto2andwithcoecientsinF;andF(F;F)denotesthevectorspaceofallmapsh:F !F.Notethatanypolynomialax2+bx+cinP2(F)mayalsoberegardedasthevectorhinF(F;F)de nedbyh( )=a 2+b +cforeach 2F.(a)IfFdenotesthesetofrealnumbers,thenshowthattwopolynono-mialsinP2(F)arelinearlyindependentinP2(F)i theyarelinearlyindependentinF(F;F).(b)IfFdenotestheintegersmod2,thenshowthattherearetwopoly-nomialsinP2(F)whichareindependentinP2(F)butarenotinde-pendentinF(F;F).(3)Insection1.6doproblems#1,2(c)(e),3(a)(e),9,14,29,31.(4)Insection2.1doproblems#1,5,6,9,11,14,281