6. Matrix with given null space

Problem. Find a matrix $M$ whose null space is the span of
$\left[\begin{array}{r}1\\ 2\\ 1\\ 2\\ 3\end{array}\right]$, $\left[\begin{array}{r}2\\ 4\\ 3\\ 7\\ 8\end{array}\right]$, and $\left[\begin{array}{r}1\\ 2\\ 3\\ 8\\ 8\end{array}\right]$.

Discussion. $M$ should have five columns; we don't yet know how many rows. Concentrate on one row of $M$; since it's unknown, call it x$= \left[\begin{array}{rrrrr}x&y&z&s&t\end{array}\right]$. When we multiply it by any of the three given column vectors we are supposed to get 0. Since x times a matrix equals x times each column of the matrix, if we let $A$ be the matrix with the given vectors as columns we have x$A = \left[\begin{array}{rrr}0&0&0\end{array}\right]$. This equation describes the space of all row vectors eligible to be a row of $M$. So we should choose $M$ to have rows that are a basis for this space. How? Just transpose to get a more familiar kind of problem, a matrix times an unknown column vector: $A^t$   x$^t =$   0. (Here we used the idea $(BC)^t = C^t B^t$.) Now we know what to do:

Method. Make $A$ with the given vectors as columns, find a basis for the null space of $A^t$, and use those basis vectors as the rows of $M$. (In doing that we're transposing the basis vectors to make them row vectors.)