$\newcommand{\Ran}{\operatorname{Ran}}$ $\newcommand{\b}{\mathbf}$If $A$ is $m\times n$ matrix and $A_{re}$ is its row reduced echelon form then I want to prove that pivot columns of $A_{re}$ forms a basis in $\Ran A_{re}$.
Let $\b v_1, ... ,\b v_p $ be the pivot columns, then by definition of pivots and reduced echelon form $\{\b v_1, ... ,\b v_p \} \subseteq \{ \b e_1, ... , \b e_m\}$ where $\b e_k$ are the standard basis of vector space $\Bbb R^m$. Therefore $\b v_1, ... ,\b v_p $ is a linearly independent system.
I need to prove that $\b v_1, ..., \b v_p$ is a generating set in $\Ran A_{re}$.
Let $\b w \in \Ran A_{re}$, $$\b w = A_{re} \b x = \sum^p_{r = 1} \alpha_r \b v_r +\sum^k_{r = 1} \beta_r \b u_r, $$where $\b u_1, ... , \b u_k$ are the columns of $A_{re}$ without pivot.
I am stuck here, how do I prove that $\b u_1, ... , \b u_k$ can be written as linear combination of $\b v_1, ... , \b v_p$ ?