Showing the existence of a bilinear map satisfying certain condition.

Question

When there is a $\mathbb{R}$-vector space $X$, can we construct a non-trivial linear map $T:X \to \mathbb{R}$ such that for some specific $x \in X$ ($x \neq 0$), $T(x) \neq 0 $?

Answer 1

Extend $\{x\}$ to a basis of $X$ and define $T(x)=1$ and ad libitum on the rest of the basis.