<s>[INST] Finding other vectors that form a basis The question I'm working on is: 
For the given vectors $v, w$ in $\mathbb R^4$, find other vectors $u_1, ..., u_n$ such that {${v, w, u_1,...,u_n}$} form a basis of $\mathbb R^4$.
$v = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}$ and $w = \begin{bmatrix} 4 \\ 3 \\ 2 \\ 1 \end{bmatrix}$
I'm not sure how I'd find other vectors that form a basis for a subspace. 
I know the first step is to get the matrix given into Reduced Row Echelon Form, and that all the vectors in a basis are linearly independent. However, putting these two together and reducing leaves:
$\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0\end{bmatrix}$
Which I'm not sure what to do with to find a basis. 
 [/INST] You want the determinant
$$
\begin{vmatrix}
1&4&a&w\\
2&3&b&x\\
3&2&c&y\\
4&1&d&z
\end{vmatrix}\ne0.
$$
Row reduction: If we subtract the previous row from each row, we get
$$
\begin{vmatrix}
1&4&a&w\\
1&-1&b-a&x-w\\
1&-1&c-b&y-x\\
1&-1&d-c&z-y
\end{vmatrix}.
$$
Repeat, and we get
$$
\begin{vmatrix}
1&4&a&w\\
0&-5&b-2a&x-2w\\
0&0&c-2b&y-2x\\
0&0&d-2c&z-2y
\end{vmatrix}
=
-5\,\begin{vmatrix}
c-2b&y-2x\\ d-2c&z-2y
\end{vmatrix}.
$$
We need, then, this last $2\times 2$ determinant to be nonzero. If we take $b=x=c=y=0$, we have 
$$
(c-2b)(z-2y)-(d-2c)(y-2x)=-dy.
$$
It is then enough to take $d=y=1$. So, 
$$
\begin{bmatrix}0\\0\\0\\1\end{bmatrix},\ \ \ \begin{bmatrix}0\\1\\0\\0\end{bmatrix}
$$
foot the bill. 
</s>