## 每週問題 March 14, 2016

Prove that the number of elements in any basis of a finite-dimensional vector space is the same as in any other basis.

$\displaystyle \mathbf{x}_1=\frac{1}{c_1}\mathbf{y}_1-\frac{1}{c_1}\sum_{i=2}^nc_i\mathbf{x}_i$

$\displaystyle \mathbf{y}_{k+1}=\sum_{i=1}^k\alpha_i\mathbf{y}_i+\sum_{i=k+1}^n\beta_i\mathbf{x}_i$