Answer to Question #311136 in Linear Algebra for Jlove

Question #311136

Which of the following sets of polynomials span P2 ?

a. {t²+1, t-1, t²+t}

B. {t²+ 2, 2t²- t+1, t + 2 , t²+t+4}


1
Expert's answer
2022-03-14T18:57:16-0400

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{c_1},\,\,{c_2},\,\,{c_3}


c1(t2+1)+c2(t1)+c3(t2+t)=0c1t2+c1+c2tc2+c3t2+c3t=0(c1+c3)t2+c2t+c1c2=0t2+0t+0\begin{array}{l} {c_1}\left( {{t^2} + 1} \right) + {c_2}\left( {t - 1} \right) + {c_3}\left( {{t^2} + t} \right) = 0\\ {c_1}{t^2} + {c_1} + {c_2}t - {c_2} + {c_3}{t^2} + {c_3}t = 0\\ \left( {{c_1} + {c_3}} \right){t^2} + {c_2}t + {c_1} - {c_2} = 0 \cdot {t^2} + 0 \cdot t + 0 \end{array}


This gives,


c1+c3=0c2=0c1c2=0{c_1} + {c_3} = 0\\ {c_2} = 0\\ {c_1} - {c_2} = 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,   2t2t+1,   t+2,  t2+t+4{{t^2} + 2,\,\,\,2{t^2} - t + 1,\,\,\,t + 2,\,\,{t^2} + t + 4} must be linearly independent.


Then for, four arbitrary constants {c_1},\,\,{c_2},\,\,{c_3},\,{c_4}\


t2+2,   2t2t+1,   t+2,  t2+t+4c1(t2+2)+c2(2t2t+1)+c3(t+2)+c4(t2+t+4)=0c1t2+2c1+2c2t2c2t+c2+c3t+2c3+c4t2+c4t+4c4=0(c1+2c2+c4)t2+(c2+c3+c4)t+(2c1+c2+2c3+4c4)=0t2+0t+0\begin{array}{l} {t^2} + 2,\,\,\,2{t^2} - t + 1,\,\,\,t + 2,\,\,{t^2} + t + 4\\ {c_1}\left( {{t^2} + 2} \right) + {c_2}\left( {2{t^2} - t + 1} \right) + {c_3}\left( {t + 2} \right) + {c_4}\left( {{t^2} + t + 4} \right) = 0\\ {c_1}{t^2} + 2{c_1} + 2{c_2}{t^2} - {c_2}t + {c_2} + {c_3}t + 2{c_3} + {c_4}{t^2} + {c_4}t + 4{c_4} = 0\\ \left( {{c_1} + 2{c_2} + {c_4}} \right){t^2} + \left( { - {c_2} + {c_3} + {c_4}} \right)t + \left( {2{c_1} + {c_2} + 2{c_3} + 4{c_4}} \right) = 0 \cdot {t^2} + 0 \cdot t + 0 \end{array}


This gives,


c1+2c2+c4=0...(1)c2+c3+c4=0...(2)2c1+c2+2c3+4c4...(3)\begin{array}{l} {c_1} + 2{c_2} + {c_4} = 0 & & ... & \left( 1 \right)\\ - {c_2} + {c_3} + {c_4} = 0 & & ... & \left( 2 \right)\\ 2{c_1} + {c_2} + 2{c_3} + 4{c_4} & & ... & \left( 3 \right) \end{array}


These are three equations, and we have four unknowns. So, there are infinite solutions.


Therefore, the are linearly dependent and hence t2+2,   2t2t+1,   t+2,  t2+t+4{{t^2} + 2,\,\,\,2{t^2} - t + 1,\,\,\,t + 2,\,\,{t^2} + t + 4}are linearly dependent and hence does not span polynomial of degree 2. 



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