(Solution) - Consider the inner product space C 0 1 with inner product -(2025 Original AI-Free Solution)

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Consider the inner product space C[0, 1] with inner product defined by

Let S be the subspace spanned by the vectors 1 and 2x - 1.
(a) Show that 1 and 2x - 1 are orthogonal.
(b) Determine ||1|| and ||2x - 1||.
(c) Find the best least squares approximation to ?x by a function from the subspace S.