Question: Define vectors in \(\mathrm{R}^{5}\) by \(v_{1}=(1,0,0,1,0), v_{2}=(2,1,3,4,2), v_{3}=(0,0,-1,3,2)\)

and \(v_{4}=(3,7,-1,3,0) .\) Let \(V \subset \mathrm{R}^{5}\) be the subspace of \(\mathrm{R}^{5}\) spanned by the vectors \(v_{1}, v_{2}, v_{3}, v_{4}\). Apply the Gram-Schmidt orthogonalisation process to this list of vectors to obtain an orthogonal basis for \(V\) with respect to the standard inner product (the dot product). If \(W=\operatorname{Span}\left(\left\{v_{1}, v_{2}\right\}\right)\) determine a basis for the orthogonal complement, \(W^{\perp}\), of \(W\) in \(V\), justifying your answer.

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