Number of variables equal to 0 in a homogeneous system of equations For an $m\times n$ matrix $A$, let $k$ be the number of variables $x_i$ in $\vec{x}=(x_1,x_2,\dots,x_n)$ for which $x_i$ must equal $0$ in the solution to $A\vec{x}=\vec{0}$. For instance the following matrix has $k=1$.
$$\begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$$
Is there anything known about $k$ (or even a name)? Is there any general technique to find $k$ for an arbitrary matrix? As far as I can see, $k\le \text{rank}(A)$ but I can't say much else beyond that.
A equivalent formulation might be, for a subspace $V\subseteq\mathbb{R}^n$, what is the maximum $k$ for which there exists a basis $B$ of $V$ such that $\{e_{i_1},e_{i_2},\dots,e_{i_k}\}\subseteq B$, where $e_{i_j}$ is a standard basis vector in $\mathbb{R}^n$. This reduces to the above when $V$ is the row space of $A$.

Actually, this idea does have a name.  One way to think about this is that a solution gives you a linear dependence relations between the columns of $A$.  The combinatorial study of these relations is formalized by matroid theory.  In matroid theory terms, one of the entries in the solution vector must be 0 if the corresponding column in the matrix is a coloop in the matroid of columns in the matrix.
One useful observation: a column is a coloop if and only if deleting it causes the rank of the matrix to drop.  That is, the last column will be a coloop if and only if when you row reduce to reduced echelon form, there's a pivot (i.e. a 1) in the last column.   For other columns, you need to move them to the end (or row reduce, saving them for last).  
