<s>[INST] Finding a new basis for the null space of a matrix Given $$
     A=\begin{pmatrix}
    -1 & -2 & 3 & -4 & -5 \\
    3 & 6 & -1 & 4 & 2 \\
    -2 & -4 & 0 & -2 & 0 \\
    -2 & -4 & 1 & -3 & 1 \\
    \end{pmatrix}
$$
and in solving for a basis for the null space of $A$, I found that:
$$ \begin{pmatrix}
 x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \\
 \end{pmatrix}=x_2 \begin{pmatrix}
 -2 \\ 1 \\ 0 \\ 0 \\ 0 \\
 \end{pmatrix} + x_4\begin{pmatrix}
 -1 \\ 0 \\ 1 \\ 1 \\ 0 \\
 \end{pmatrix}$$
Thus, a basis for the null space of A is $$B_A=\begin{Bmatrix} \begin{pmatrix}
 -2 \\ 1 \\ 0 \\ 0 \\ 0 \\
 \end{pmatrix},\begin{pmatrix}
 -1 \\ 0 \\ 1 \\ 1 \\ 0 \\
 \end{pmatrix} \end{Bmatrix} $$ 
My question is, how would I find a new basis for the null space of $A$, whose vectors are NOT multiples?  Are there any fast ways of doing this?
I am aware that you can use a change of basis matrix, but if a question asks you to find this new basis for null space and then find the change of basis matrix (not the other way around), how would this be done?
 [/INST] You probably want to find an orthonormal basis.  Look into the Gram-Schmidt process.
</s>