5 linear algebra question1. Let V be an inner product space and let W be a finite-dimensional subspace of V. If x∉Wx∉W prove that there exists y∈Vy∈V such that y∈W⊥y∈W⊥ but ⟨xy⟩≠0〈x y〉≠0. 2. Let ββ be a basis for a subspace W of an inner product space V and let z∈Vz∈V. Prove that z∈W⊥z∈W⊥ if and only if ⟨zv⟩=0〈z v〉=0 for every v∈βv∈β. 3. Let V be an inner product space S and S0S0 be subsets of V and W be a finite-dimensional subspace of V. Prove the following results. (a)S0⊆SS0⊆S implies that S⊥⊆S⊥0S⊥⊆S0⊥. (b)S⊆(S⊥)⊥;S⊆(S⊥)⊥; so span(S)⊆(S⊥)⊥span(S)⊆(S⊥)⊥. (c)W=(W⊥)⊥W=(W⊥)⊥. (d)V=W⊕W⊥V=W⊕W⊥. 4. Let V be a finite-dimensional inner product space and let T be a linear operator on V. Prove that if T is invertible then T* is invertible and (T*)−1=(T−1)*(T*)−1=(T−1)*. 5. Let V be an inner product space and let T be a linear operator on V. Prove the following results. (a)R(T*)⊥=N(T).