PROBLEM: Use as many columns of $A$ as possible to construct a matrix $B$ with the property that the equation $B x = 0$ has only the trivial solution.
Thoughts: Since I am doing this problem using Mathematica, I can pick and choose certain columns that will give me a matrix that reduces to Reduced Echelon Form. I chose columns $X_1$, $X_2$, $X_4$, and $X_5$. This new matrix $B$ row reduces completely, such that there is a pivot in each column.
Do I have the right idea? Is this what the question asked me to do?
What you did is correct. You are searching for a matrix with kernel/null space equal to $\{0\}$. Here is the idea behind your answer.
As this cannot hold for five columns, because this means there will be a linearly dependent column, you must remove a linearly dependent column.
Now you found a matrix that has reduced echelon form $I$, which means it is invertible, thus bijective, so only $x=0$ is the solution to $Bx=0$.