# Thread: Find basis for A

1. ## Find basis for A

For matrix $A=\begin{pmatrix}1&-1&4&-6&8&14\\-1&1&-3&5&-6&-10\\2&-2&6&-10&12&21\\-1&1&-2&4&-4&-8\\0&0&-1&1&-2&1\end{pmatrix}$

I found $rref(A)=\begin{pmatrix}1&-1&0&-2&0&0\\0&0&1&-1&2&0\\0&0&0&0&0&1\\0&0&0&0&0&0\\0&0&0&0&0&0\end{p matrix}$

x1-x2-2x4=0
x3-x4+2x5=0
x6=0

x1=x2+2x4
x2=x1-2x4
x3=x4-2x5
x4=x3+2x5
x5=(x4-x3)/2
x6=0

Now I don't know what to do because there are so many elements. Thank you in advance for any help you can give me.

2. Could I do this?
$
x_{1}=1x_{2}+2x_{4}
x_{3}=1x_{4}-2x_{5}
x_{6}=0
$

$c_{2}[1,1,0,0,0,0]+c_{4}[2,0,1,1,0,0]+c_{5}[0,0,-2,0,1,0]$

Then I would have basis
$(c_{2}[1,1,0,0,0,0], c_{4}[2,0,1,1,0,0],c_{5}[0,0,-2,0,1,0])$