## 每週問題 October 17, 2016

Let $\mathbf{a}_1,\ldots,\mathbf{a}_n$ be $n$ vectors in $\mathbb{R}^m$. Show that $m\times m$ Gramian matrix $G=\sum_{i=1}^n\mathbf{a}_i\mathbf{a}_i^T$ is nonsingular if and only if $\hbox{span}\{\mathbf{a}_1,\ldots,\mathbf{a}_n\}=\mathbb{R}^m$.

$m\times n$ 階矩陣 $A=\begin{bmatrix} \mathbf{a}_1&\cdots&\mathbf{a}_n \end{bmatrix}$。寫出

$\displaystyle G=\sum_{i=1}^n\mathbf{a}_i\mathbf{a}_i^T=\begin{bmatrix} \mathbf{a}_1&\cdots&\mathbf{a}_n \end{bmatrix}\begin{bmatrix} \mathbf{a}_1^T\\ \vdots\\ \mathbf{a}_n^T \end{bmatrix}=AA^T$