Determining standard basis Suppose $s=\{u_1,u_2,u_3\}$ is a basis for a subspace $V \subset \mathbb{R}^4$. Let $v$ be a vector in $V$. Suppose that the RREF of $\{u_1,u_2,u_3\mid v\}$ is
$$\left[\begin{array}{ccc|c}1&0&0&1\\0&1&0&-2\\0&0&1&0\\0&0&0&0\end{array}\right]$$
What is $(v)_s$ (standard basis)??

It seems clear that this should be $(1,-2,0)$ as the fourth column in the RREF equals the first column (corresponding to $u_1$) plus -2 times the second (corresponding to $u_2$). Such a relation is preserved (both ways) by all reversible row operations and holds in the final form, so held at the outset too. So $v = u_1 - 2u_2$ and hence this form.
