Let $\mathcal{M}$ be the set of $5 \times 5$ real matrices of rank 3 . Given a matrix $A$ in $\mathcal{M}$, the set of columns of $A$ has $2^{5}-1=31$ nonempty subsets. Let $k_{A}$ be the number of these subsets that are linearly independent.

Determine the maximum and minimum values of $k_{A}$, as $A$ varies over $\mathcal{M}$.
The rank of a matrix is the dimension of the span of its columns.
The maximum possible value of $k_{A}$ is 25 , and the minimum possible value is 7 .
Let $A \in \mathcal{M}$. Then since $\operatorname{rank}(A)=3$ a linearly independent subset of the set of columns of $A$ has size at most 3 , so
$$
k_{A} \leq\binom{ 5}{1}+\binom{5}{2}+\binom{5}{3}=5+10+10=25 .
$$

Now let $S$ be a linearly independent subset of the columns of $A$ of size 3 . Then every nonempty subset of $S$ is also linearly independent, so
$$
k_{A} \geq 2^{3}-1=7 \text {. }
$$

We conclude that $7 \leq k_{A} \leq 25$.
To complete the proof let
$$
A_{1}=\left[\begin{array}{lllll}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{array}\right], \quad A_{2}=\left[\begin{array}{lllll}
1 & 0 & 0 & 1 & 1 \\
0 & 1 & 0 & 1 & 2 \\
0 & 0 & 1 & 1 & 3 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{array}\right] .
$$

Then $A_{1}, A_{2} \in \mathcal{M}$ and we have $k_{A_{1}}=7, k_{A_{2}}=25$, showing that these bounds are sharp.