“Let $V$ be a finite-dimensional vector space with basis $\{v_1,\ldots,v_n\}$ and let $\{r_1,\ldots,r_n\}$ be a set of $n$ scalars. There exists one and only one linear functional $L\colon V\longrightarrow\mathbb R$ such that: $L(v_i) = r_i$ for every $i$”
Could you help me understand this lemma on linear functionals with one example?
For instance, supppose that $V=\{(x,y,z)\in\mathbb{R}^2\,|\,x+y+z=0\}$. Consider the basis $\{(1,-1,0),(0,1,-1)\}$ of $V$ and the numbers $1$ and $-1$. Is there a linear functional $L\colon V\longrightarrow\mathbb R$ such that $L(1,-1,0)=1$ and that $L(0,1,-1)=-1$? Yes, there is. Just take $L(x,y,z)=x+z$.