1.9.6 Linear spaces

Problem: Prove that the set of vectors \( L=\left\{\bar{a}=\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}\right) \mid \alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}=0\right\} \) is the subspace of the space \( R^{n} \), a sequence of \( n \)-dimensional vectors \( \bar{a}_{1}=(1,0, \ldots, 0,-1) \), \( \bar{a}_{2}=(0,1, \ldots, 0,-1), \ldots, \bar{a}_{n-1}=(0,0, \ldots, 1,-1) \quad \) basis of this subspace.