## 每週問題 May 8, 2017

Let $\mathbf{v}_1,\ldots,\mathbf{v}_{n+1}$ be vectors in $\mathbb{R}^n$ ($n\ge 2$) such that $\mathbf{v}_i^T\mathbf{v}_j<0$ for $i\neq j$. Prove that any $n$ of these vectors form a basis of $\mathbb{R}^n$.

$\displaystyle 0=\mathbf{v}_{n+1}^T\left(\sum_{i=1}^kc_i\mathbf{v}_i\right)=\sum_{i=1}^kc_i\mathbf{v}_{n+1}^T\mathbf{v}_i$

$\displaystyle \left(\sum_{i=1}^pc_i\mathbf{v}_i\right)^T\left(\sum_{j=p+1}^kc_j\mathbf{v}_j\right)=\sum_{i=1}^p\sum_{j=p+1}^kc_ic_j\mathbf{v}_i^T\mathbf{v}_j\ge 0$

$\displaystyle \left(\sum_{i=1}^pc_i\mathbf{v}_i\right)^T\left(\sum_{j=p+1}^kc_j\mathbf{v}_j\right)=-\left(\sum_{i=1}^pc_i\mathbf{v}_i\right)^T\left(\sum_{i=1}^pc_i\mathbf{v}_i\right)\le 0$

### 3 Responses to 每週問題 May 8, 2017

1. Wenchao Deng 說道：

Reblogged this on I think therefore I am. and commented:
interesting

2. 林源倍 說道：

哇，很不錯的題目

• ccjou 說道：

歡迎林老師大駕光臨