Let \(A\) be the matrix with columns the above vectors: \[\begin{bmatrix} 2 & 2 \\ 4 & 1 \\ -8 & -8 \\ 6 & 3 \end{bmatrix}\] Reduced row echelon form (RREF) of \(A^T\): \[\begin{bmatrix} 1 & 0 & -4 & 1 \\ 0 & 1 & 0 & 1 \end{bmatrix}\] Basis for \(W^\perp = (\textrm{Col}A)^\perp = \textrm{Nul} A^T\): \[\begin{bmatrix} 4 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \qquad \begin{bmatrix} -1 \\ -1 \\ 0 \\ 1 \end{bmatrix}.\]

Problem:

Let \(W\) be the subspace spanned by the vectors    
\[
\begin{bmatrix} 
 2  \\ 
 4  \\ 
 -8  \\ 
 6
\end{bmatrix}, \qquad 
\begin{bmatrix} 
 2  \\ 
 1  \\ 
 -8  \\ 
 3
\end{bmatrix}.\]
Find a basis for the orthogonal complement \(W^\perp\) of \(W\).

\begin{solution}
Solution:

Let \(A\) be the matrix with columns the above vectors:
\[\begin{bmatrix} 
2 & 2   \\ 
4 & 1   \\ 
-8 & -8   \\ 
6 & 3
\end{bmatrix}\]
Reduced row echelon form (RREF) of \(A^T\):
\[\begin{bmatrix} 
1 & 0 & -4 & 1   \\ 
0 & 1 & 0 & 1
\end{bmatrix}\]
Basis for \(W^\perp = (\textrm{Col}A)^\perp = \textrm{Nul} A^T\):   
\[\begin{bmatrix} 
 4  \\ 
 0  \\ 
 1  \\ 
 0
\end{bmatrix}, \qquad 
\begin{bmatrix} 
 -1  \\ 
 -1  \\ 
 0  \\ 
 1
\end{bmatrix}.\]
\end{solution}