每週問題 March 21, 2016

If $\mathcal{V}$ is a finite-dimensional vector space and if $\{\mathbf{y}_1,\ldots,\mathbf{y}_m\}$ is any set of linearly independent vectors in $\mathcal{V}$, prove that, unless $\{\mathbf{y}_1,\ldots,\mathbf{y}_m\}$ already form a basis, we can find vectors $\mathbf{y}_{m+1},\ldots,\mathbf{y}_{m+p}$ so that $\{\mathbf{y}_1,\ldots,\mathbf{y}_m,\mathbf{y}_{m+1},\ldots,\mathbf{y}_{m+p}\}$ is a basis.

$S=\{\mathbf{y}_1,\ldots,\mathbf{y}_m,\mathbf{x}_1,\ldots,\mathbf{x}_n\}$

$S'=\{\mathbf{y}_1,\ldots,\mathbf{y}_m,\mathbf{x}_1,\ldots,\mathbf{x}_{i-1},\mathbf{x}_{i+1},\ldots,\mathbf{x}_n\}$