Let $A=(A_1,A_2,A_3,A_4,A_5)$ be a $4\times 5$ matrix. Assume that the general solution is given by $X=(x_1,2x_1-x_3,x_3,x_3+x_1,2x_3)^T$.
(a) Find a maximal independent subset of $\{A_1,A_2,A_3,A_4,A_5\}.$
(b) Show that $A_1+A_2+A_3+A_4+A_5$ is a linear combination of the maximal independent subset.
At first, I thought about the old way I was taught to find a maximally independent subset where only the rows with a pivot are considered. In this case, it would be $x_1$ and $x_3$.
I'm not sure how to work one of these problems backwards - that is, where you are given the solution $X$ and need to find a maximal independent subset.
The fact that the free variables are the first and the third means that the columns $A_2,A_4,A_5$ form a basis for the column space. This answers part (a).
As for part (b), $\{A_2,A_4,A_5\}$ being a basis of $C(A)$ implies that any linear combination of the columns may be reduced to a combination of these three alone.