Let T : P3 -> R
T(a0 + a1x + a2x^2 + a3x^3) = a0 - a1 + a2 - a3
Find the Rank and Nullity of T and find a basis for the kernel of T.
For my work :
I set a3 = a0 - a1 + a2
Plugged this into a3 for the left hand side and after some simplifications, I get
a0(1 + x^3) +a1(x - x^3) + a2(x^2 + x^3)
Basis = [ (1 + x^3), (x - x^3), (x^2 + x^3) ]
Now my professor says the nullity is 3 which makes the rank = 1 but i'm not understanding WHY it's 3. Also, my professor chose initially to set a3 = a0 - a1 + a2. If I had chosen to set say a1 = a0 + a2 - a3 would it still be correct even though the Basis would look different?