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(Solved): Let U be a subset of an inner product space V. The orthogonal complement of U. denoted , is the set ...


Let U be a subset of an inner product space V. The orthogonal complement of U. denoted U^{\perp }, is the set of all vectors in V that are orthogonal to every vector in U. That is,

U^{\perp }=\left \{ \vec{v}\epsilon V:<\vec{v},\vec{u}.=0 for all \vec{u}\epsilon U \right \}. Show that U^{\perp } is always a subspace of V. Also show that V^{\perp }=\left \{ \vec{0} \right \} and \left \{ \vec{0} \right \}^{\perp }=V



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