Find the orthogonal projection of the function $f(x) = x$ onto the subspace of $C([-1,1])$ spanned by $S = \{\sin(\pi x), \sin(2\pi x), \sin(3\pi x)\}$
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The inner product of two functions f(x) and g(x) over the interval [-1,1] is defined as: ⟨f,g⟩ = ∫[a,b] f(x)g(x) dx In this case, we need to find the inner product of f(x) with each basis vector in S. Let's start with sin(Tx): ⟨f, sin(Tx)⟩ = ∫[-1,1] x * Show more…
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