Question

write the polynomial V = t^2+4t-3 over R as a linear combination of the polynomials a=t^2-2t+5, b=2t^2-3t and c=t+3.appreciate ur help

Answer 1

i am not sure but
u have to choose coefficient beforre a,b,c to present v=a1a+a2b+a3c
they are unknown and u should find them write a1(t^2-2t+5)+a2(2t^2-3t)+a3(t+3) this should be equal to t^2+4t-3 then u should bare the brackets and group terms t^2 ,t, free therm a1(t^2-2t+5)+a2(2t^2-3t)-a3(t+3)
then u have
t^2(a1+2a2)-t(3a2+2a1+a3)+3+5a1
next step is to equal each therm
t^2(a1+2a2)-t(3a2+2a1+a3)+3+5a1=t^2+4t-3
a1+2a2=1
3a2+2a1+a3=4
3+5a1=-3
and u have liner system
solve it and a1=-6/5
a2=11/10
a3=27/5
then linear combination is -6/5a+11/10b+27/5c

Answer 2

I can miscount something

Answer 3

\[(xt ^{2}+y2t ^{2})+(-2xt-3yt+zt)+(5x+3z)=t ^{2}+4t-3\]
\[(x+2y)t ^{2}+(-2x-3y+z)t+(5x+3z)=t ^{2}+4t-3\]


therefore x=-3 ; y=2 ;z=4

checking
lhs=rhs