Solution (a)
If \left\{ {{t^2} + 1,\,\,t - 1,\,\,{t^2} + t} \right\}\ , span P2, then {t^2} + 1,\,\,t - 1,\,\,{t^2} + t\ must be linearly independent.
Then for, three arbitrary constants c1,c2,c3
c1(t2+1)+c2(t−1)+c3(t2+t)=0c1t2+c1+c2t−c2+c3t2+c3t=0(c1+c3)t2+c2t+c1−c2=0⋅t2+0⋅t+0
This gives,
c1+c3=0c2=0c1−c2=0
We solve and get {c_1} = {c_2} = {c_3} = 0\
Therefore, {t^2} + 1,\,\,t - 1,\,\,{t^2} + t\are linearly independent and hence span polynomial of degree 2.
Solution (b)
If, \left\{ {{t^2} + 2,\,\,\,2{t^2} - t + 1,\,\,\,t + 2,\,\,{t^2} + t + 4} \right\}\ span P2, then t2+2,2t2−t+1,t+2,t2+t+4 must be linearly independent.
Then for, four arbitrary constants {c_1},\,\,{c_2},\,\,{c_3},\,{c_4}\
t2+2,2t2−t+1,t+2,t2+t+4c1(t2+2)+c2(2t2−t+1)+c3(t+2)+c4(t2+t+4)=0c1t2+2c1+2c2t2−c2t+c2+c3t+2c3+c4t2+c4t+4c4=0(c1+2c2+c4)t2+(−c2+c3+c4)t+(2c1+c2+2c3+4c4)=0⋅t2+0⋅t+0
This gives,
c1+2c2+c4=0−c2+c3+c4=02c1+c2+2c3+4c4.........(1)(2)(3)
These are three equations, and we have four unknowns. So, there are infinite solutions.
Therefore, the are linearly dependent and hence t2+2,2t2−t+1,t+2,t2+t+4are linearly dependent and hence does not span polynomial of degree 2.
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