Q:

help with span/linearindepencence/basis

Determine whether each given tuple (i) spans $P_2$, (ii) is linearly independent, (iii) is a basis of $P_2$.
(a.) $(2−x, 1+3x−x^2 , −7x+2x^2 )$
(b.) $(1+x−x^2 , 3x+2x^2 , −1+x+4x^2 )$
I can't figure this one out.
Thanks for the help!!

A:

Since $dim(P_2)=3$, any three linearly independent vectors would span the whole set. All it remains is to show that which of the following is a linearly independent set. 
For (i)  if $c_1(2-x)+c_2(1+3x-x^2)+c_3(-7x+2x^2)=0$, then $2c_1+c_2=0,-c_1+3c_2-7c_3=0,2c_3-c_2=0$. Now if you take $c_1=-\frac{1}{2},c_2=1, c_3=\frac{1}{2}$, you will see that there exists a nonzero linear combination of the three vectors which equals zero. Hence the set is not linearly independent and they don't form a basis.
Similarly check for (ii).

