Linear Functionals Respect Differences, and Coefficients Along a Nonzero Vector Are Unique

In the setting of Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity and Real Normed, Inner Product and Hilbert Spaces: Notation, let $V$ be a real vector space.

(i) Differences and functionals. Let $W$ be a linear subspace of $V$, let $g$ be a linear functional on $W$, and let $u,v\in W$. Then $u-v\in W$ and

$$g(u-v)=g(u)-g(v).$$

(ii) Coefficients along a nonzero vector. Let $x_{0}\in V$ with $x_{0}\neq0_{V}$, and let $t,t'\in\mathbb{R}$. If $tx_{0}=t'x_{0}$, then $t=t'$.