4. Linear relations between vectors

Problem. Find a basis for the linear relations between these vectors:

$\left[\begin{array}{r}1\\ 2\\ 1\end{array}\right]$, $\left[\begin{array}{r}2\\ 4\\ 2\end{array}\right]$, $\left[\begin{array}{r}1\\ 3\\ 3\end{array}\right]$, $\left[\begin{array}{r}2\\ 7\\ 8\end{array}\right]$, $\left[\begin{array}{r}3\\ 8\\ 8\end{array}\right]$

Method. A linear relation means a linear combination of these vectors that equals 0. Treat the list of coefficients as a vector itself. In other words, we are looking for a basis for the space of coefficients

$\left[\begin{array}{r}x\\ y\\ z\\ s\\ t\end{array}\right]$ such that $x \left[\begin{array}{r}1\\ 2\\ 1\end{array}\right] + y \left[\begin{array}{... ...8\\ 8\end{array}\right] = \left[\begin{array}{r}0\\ 0\\ 0\end{array}\right]$

As in Section , this is the same as saying $M$x$=$   0, where $M$ is the matrix with these columns, so the answer is a basis for the null space of $M$.