(Doing part (ii) of this question)
What i did was find the wronskian determinant:
$$ \begin{matrix} 1 & t-t^2 \\ 0 & 1-2t \\ \end{matrix} $$
So determinant is 1(1-2t) = 1- 2t which does not equal zero, therefore subset is linearly independant.
Would this be the correct method?
With respect to the standard basis $\{1,t,t^2\}$ the given vectors have coordinates
$1=1\cdot 1+0\cdot t + 0 \cdot t^2$ that is $$(1,0,0)$$
$t(1-t)=t-t^2=0\cdot 1+1\cdot t + (-1) \cdot t^2$ that is $$(0,1,-1)$$
and they are linearly independent.