Get solution: For Exercises 3742, u and v (and their subscripted relatives) arevectors

Chapter 10, Problem 42

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For Exercises 3742, u and v (and their subscripted relatives) arevectors in an inner product space V, and S is a nonzero finitedimensionalsubspace of V.Here we prove that the GramSchmidt process works. Supposethat {u1, ... , uk } are linearly independent vectors, and that{v1, ... , vk } are as defined in the statement of the GramSchmidtprocess.(a) Use induction to show {v1, ... , v j} is an orthogonal set forj = 1, ... , k.(b) Use induction to show span{u1, ... , uj} = span{v1, ... , v j}for j = 1, ... , k.(c) Explain why (a) and (b) imply that GramSchmidt yields anorthogonal basis.