## 每週問題 April 7, 2014

Define a non-zero linear functional $f$ on $\mathbb{C}^3$ such that if $\mathbf{u}=(1,1,1)$ and $\mathbf{v}=(1,1,-1)$, then $f(\mathbf{u})=f(\mathbf{v})=0$.

$\displaystyle f(x_1,x_2,x_3)=c_1x_1+c_2x_2+c_3x_3$

$(1,1,1)$$(1,1,-1)$ 代入上式，

\displaystyle\begin{aligned} c_1+c_2+c_3&=0\\ c_1+c_2-c_3&=0. \end{aligned}