Let \(W\) be the subspace spanned by the vectors \[ \begin{bmatrix} 0 \\ -2 \\ -6 \\ -4 \end{bmatrix}, \qquad \begin{bmatrix} -5 \\ -3 \\ -34 \\ -26 \end{bmatrix}.\] Find a basis for the orthogonal complement \(W^\perp\) of \(W\).
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Let \(W\) be the subspace spanned by the vectors \[ \begin{bmatrix} 0 \\ -2 \\ -6 \\ -4 \end{bmatrix}, \qquad \begin{bmatrix} -5 \\ -3 \\ -34 \\ -26 \end{bmatrix}.\] Find a basis for the orthogonal complement \(W^\perp\) of \(W\).
\begin{solution} Solution: Let \(A\) be the matrix with columns the above vectors: \[\begin{bmatrix} 0 & -5 \\ -2 & -3 \\ -6 & -34 \\ -4 & -26 \end{bmatrix}\] Reduced row echelon form (RREF) of \(A^T\): \[\begin{bmatrix} 1 & 0 & 5 & 4 \\ 0 & 1 & 3 & 2 \end{bmatrix}\] Basis for \(W^\perp = (\textrm{Col}A)^\perp = \textrm{Nul} A^T\): \[\begin{bmatrix} -5 \\ -3 \\ 1 \\ 0 \end{bmatrix}, \qquad \begin{bmatrix} -4 \\ -2 \\ 0 \\ 1 \end{bmatrix}.\] \end{solution}