Finding Kernel of linear transformation

Question

Let T:R3→R2 be the linear transformation defined by

T(x,y,z)=(x−y−2z,2x−2z)

Then Ker(T) is a line in R3, written parametrically as

r(t)=t(a,b,c) for some (a,b,c)∈R3

(a,b,c) = . . .

(Write your answer as a vector (a,b,c). For example "(2,3,4)")

So, when I tried to solve the question making x-y-2z = 0, and 2x-2z= 0 I got the answer x(1,1,-2) but it seems that it's incorrect. Can someone help me with this?

Answer 1

Because $x=z$ gives $y=-z$ and we obtain $$\ker(T)=\{t(1,-1,1)|t\in\mathbb R\}.$$