Hiu Nga--Cheung>>hi TA Bong_Lee>>hi TA Bong_Lee>>any q? Hiu Nga--Cheung>>i have q Hiu Nga--Cheung>>on discussion board TA Bong_Lee>>wait TA Bong_Lee>>now i am answering your last q on discussion board TA Bong_Lee>>T(p(t) is linear transformation map polynomial of degree2 to itself TA Bong_Lee>>as we know every polynomial of deg2 can be represented as ax^2+bx+c TA Bong_Lee>>with two of above fact TA Bong_Lee>>T(ax^2+bx+c )=T(ax^2)+T(bx)+T(c) TA Bong_Lee>>=aT(x^2)+bT(x)+cT(1) *-**** Siu--Him--Lau left MATH113_Room1. Time: Mon Oct 21 22:19:05 2002 TA Bong_Lee>>so the range is spaned by T(x^2),T(x),T(1) TA Bong_Lee>>see? Hiu Nga--Cheung>>why T(1)=1+t TA Bong_Lee>>do u know the meaning of T(p(t)) = p(0) + p(1)t + (p'(0) + p'(1))t2? Hiu Nga--Cheung>>no TA Bong_Lee>>it means u take a polynomial of deg 2 name it p(t) TA Bong_Lee>> then u put t=1 TA Bong_Lee>>so u get p(1) TA Bong_Lee>>and u put t=0 TA Bong_Lee>>so u get p(0) TA Bong_Lee>>and u also take the first derivative p'(t) TA Bong_Lee>>put t=1 and t=0 in it TA Bong_Lee>>so u get p'(1) and p'(0) TA Bong_Lee>>now use the value p(0),p(1),p'(0) and p'(1) to construct another polynomial TA Bong_Lee>>for p(t)=1 TA Bong_Lee>>p(0)=p(1)=1 TA Bong_Lee>>and the first derivative=0 TA Bong_Lee>>so T(1)=1+t TA Bong_Lee>>get it? Hiu Nga--Cheung>>why for p(t)=1 p(0)=p(1)=1 TA Bong_Lee>>p(t)=1 means the polynomial p(t) equal 1for all t TA Bong_Lee>>it's a constant function Hiu Nga--Cheung>>oh Hiu Nga--Cheung>>then why T(t)=t+2t^2 TA Bong_Lee>>u still don't know the meaning of TA Bong_Lee>> T(p(t)) = p(0) + p(1)t + (p'(0) + p'(1))t2? TA Bong_Lee>>u can let p(t)=t and try Hiu Nga--Cheung>>for p(t)=1 p(0)=p(1)=1 then for p(t)=t p(0)=1? p'(0)=1? TA Bong_Lee>>for p(t)=t p(0)=0 TA Bong_Lee>>p'(0)=1 is correct Hiu Nga--Cheung>>p(t)=t^2 then p(0)=0, p(1)=1, p'(0)=0, p'(1)=2 TA Bong_Lee>>correct!u get it!congradulations! Hiu Nga--Cheung>>next q then TA Bong_Lee>>on diss board? Hiu Nga--Cheung>>yes TA Bong_Lee>>in fact col5 can be spaned by col 1 2 and 4 Hiu Nga--Cheung>>i just see col5 is linear combination of col3 ,col4, TA Bong_Lee>>yes TA Bong_Lee>>but it doesn't make any contradiction! TA Bong_Lee>>we can use all col 1,2,3,4, to represent col5 Hiu Nga--Cheung>>but we cannot use col 1,2,4 to represent col3? TA Bong_Lee>>yes we can TA Bong_Lee>>just times col4 by 0 Hiu Nga--Cheung>>in notes :The nonpivot columns of A are linear combinations of the preceeding columns Hiu Nga--Cheung>>what is preceeding columns TA Bong_Lee>>col on it's left hand side TA Bong_Lee>>ok let 's finish here TA Bong_Lee>>bye