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[See Solution] Equip R^4 with the Euclidean inner product (the dot product.) Let W be the subspace spanned by v_1=(1,0,1,1), v_2=(-1,1,0,1), v_3=(1,-1,1,0)

Question: Equip \(\mathbb{R}^{4}\) with the Euclidean inner product (the dot product.) Let \(W\) be the subspace spanned by

\[\mathbf{v}_{1}=(1,0,1,1), \mathbf{v}_{2}=(-1,1,0,1), \mathbf{v}_{3}=(1,-1,1,0)\]
  1. Find an orthonormal basis for \(W\).
  2. Find the element of \(W\) that is closest to \((1,1,1,1)\).
  3. Find a basis for \(W^{\perp}\).
  4. Find the element of \(W\) that is closest to \((0,8,8,-8)\).
  5. Find the matrix of the projection \(P_{W}\) with respect to the standard basis of \(\mathbb{R}^{4}\).

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Solution: The downloadable solution consists of 4 pages
Deliverable: Word Document


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