## 每週問題 October 15, 2012

Let $A$ be an $n\times n$ matrix and $\mathbf{v}_1,\ldots,\mathbf{v}_n\in\mathbb{C}^n$ be linearly independent. Show
that $A$ is nonsingular if and only if $A\mathbf{v}_1,\ldots, A\mathbf{v}_n$ are linearly independent.

$A$ 是可逆矩陣，考慮

$c_1A\mathbf{v}_1 +\cdots + c_nA\mathbf{v}_n = \mathbf{0}$

$A(d_1\mathbf{v}_1 +\cdots + d_n\mathbf{v}_n) = d_1A\mathbf{v}_1 +\cdots + d_nA\mathbf{v}_n = \mathbf{0}$

PowSol-Oct-15-12