I have a matrix $A \in \mathbb{R}^{n \times p}$ where $p>n$ and column rank of $A=n$. Consider a matrix $M=[e \quad A^T]$ where $e \in \mathbb{R}^p$ is a vector of $1$. What is the column rank of $M$ ?
If $M$ doesn't have full column rank, then can we find certain columns of $A$ deleting which, will lead to a new full column rank matrix $\hat{M}=[e \quad \hat{A}^T]$, where $\hat{A}$ is obtained by deleting certain columns of $A$. However, I want to delete such potentials columns, as few as possible.