$T (x_1,x_2,x_3,...,x_n) = (-x_3,x_3,x_4,x_5,...) $ then $ W \ne ker T$

Question

Let $V$ the vector space of all sequences of real numbers and $W$ the subspace given by $W = \{(a,a,0,0,...) | a \in R\}$ , and $T : V \rightarrow V$ given by $T (x_1,x_2,x_3,...,x_n) = (-x_3,x_3,x_4,x_5,...) $ then $ W \ne ker T$ .

How can i prove this?, some help please.