<s>[INST] How to decompose $X_p = X_r +X_n$? From an original matrix $A$, I have solved for $X_p$, $C(A^T)$, and $N(A)$. The last step is the problem below. 
How should I approach this? Thank you for taking the time. 
Decompose $X_p = X_r + X_n$, where $X_r \in C(A^T)$ and $X_n \in N(A)$.
$$
X_p =    \begin{pmatrix}
    3 \\
    4 \\
    -2 \\
    0 \\
    \end{pmatrix}, \quad
C(A^T) = \text{span}\left[\begin{pmatrix}16\\2\\3\\13\end{pmatrix},\begin{pmatrix}5\\11\\10\\8\end{pmatrix},\begin{pmatrix}9\\7\\6\\12\end{pmatrix}\right], \quad
N(A) = \begin{bmatrix}-1\\-3\\3\\1\end{bmatrix}$$
 [/INST] Since I am breaking my Xp into orthogonal projections, I would just need to solve for my scalars at this point (X below).AX=B solving for X. So my final solution would be my C(AT) and N(A) multiplied by my scalar (x).
A = \begin{pmatrix}
    16,5,9,-1 \\
    2,11,7,-3 \\
    3,10,6,3 \\
    13,8,12,1 \\
    \end{pmatrix}
B = \begin{pmatrix}
    3 \\
    4 \\
   -2 \\
    0 \\
    \end{pmatrix}
X = \begin{pmatrix}
    159/85 \\
    1909/1360 \\
   -3853/1360 \\
    -31/20 \\
    \end{pmatrix}
</s>