例题 3.1

Question

设 ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n$ 为向量空间 ${\mathbb{R}}^n$ 的任意一组标准正交基, $\alpha \in {\mathbb{R}}^n,\alpha =k_1{\alpha }_1+k_2{\alpha }_2+\cdots +k_n{\alpha }_n$, 证明: $k_i=\left({\alpha }_i,\alpha \right),i=1,2,\cdots ,n$.

证明

用 ${\alpha }_i\left(i=1,2,\cdots ,n\right)$ 与 $\alpha$ 作内积,由于 $\left({\alpha }_i,{\alpha }_j\right)=0\left(i\ne j\right),\left({\alpha }_i,{\alpha }_i\right)=$ 1, 得到

$$\begin{array}{rl}\left({\alpha }_i,\alpha \right)& =\left({\alpha }_i,k_1{\alpha }_1+k_2{\alpha }_2+\cdots +k_n{\alpha }_n\right)\\ & =\left({\alpha }_i,k_i{\alpha }_i\right)=k_i\left({\alpha }_i,{\alpha }_i\right)=k_i.\end{array}$$