Getting linear combinations in linear algebra?

Question

I failed a homework problem a few days ago. I can't figure out how they got the answers, which have been given in green as corrections. Help me figure how they got them;

Answer 1

Example. Here is how you do the second one, see if you can do the others for yourself.

We have $$f_2(t)=1+t=a_0+a_1t+a_2t^2\quad\hbox{where}\quad a_0=1\,,\ a_1=1\,,\ a_2=0\,.$$ Substituting into the rule given for $T$, we have $$\eqalign{T(f_2(t)) &=(2a_0+4a_1+4a_2)+(2a_0+2a_1+3a_2)t+(6a_0+4a_1+4a_2)t^2\cr &=6+4t+10t^2\cr &=2(1)-6(1+t)+10(1+t+t^2)\qquad\qquad\qquad\qquad(*)\cr &=2f_1(t)-6f_2(t)+10f_3(t)\ .\cr}$$

Now to explain how I got line $(*)$: we need $$6+4t+10t^2=\lambda_1f_1(t)+\lambda_2f_2(t)+\lambda_3f_3(t)\ ,$$ that is, $$6+4t+10t^2=\lambda_1(1)+\lambda_2(1+t)+\lambda_3(1+t+t^2)\ .$$ Expand and equate coefficients: $$\lambda_1+\lambda_2+\lambda_3=6\ ,\quad \lambda_2+\lambda_3=4\ ,\quad \lambda_3=10\ .$$ I am sure you can now solve these to find $\lambda_1,\lambda_2,\lambda_3$.