Question

Will give medal .
Compute a basis matrix for the null space of the matrix A and express the points \[x_i\] as \[x_i=p_i+q_i\] where \[p_i\] is the null space of A and \[q_i\] is the range space of \[A^T\]. \[A=\left[\begin{matrix}1 & 1& 1 & 1 \\ 2 & 0 & 0 & 2\\ 1& -1 & -1 & 1\end{matrix}\right]\] \[x_1=\left(\begin{matrix}3 \\ 1\\ 1\\ 2\end{matrix}\right)\] \[x_2=\left(\begin{matrix}8 \\ 9\\ -2\\ -4\end{matrix}\right)\]

Answer 1

For my null space I got \[p=\left(\begin{matrix}v_1 \\ v_2\\ -v_2\\ -v_1\end{matrix}\right)\] From which I constructed a basis matrix \[z=\left[\begin{matrix}1 & 0\\ 0 & 1\\ 0 & -1\\ -1 & 0\end{matrix}\right]\] For the range space I got after taking the transpose of the matrix A \[q=\left(\begin{matrix}v_3 \\ v_4\\ v_4\\ 0\end{matrix}\right)\]. I think my range space is wrong and I'm not sure how to do the equation of x=p+q as a result.