(Solution) - Let x1 x2 xn be a distinct -(2025 Original AI-Free Solution)

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Let x1, x2,. . . , xn be a distinct set of sample points.
(a) Prove that the functions f1(x),... , fk(x) are linearly independent if their sample vectors f1......... f? are linearly independent vectors in Rn.
(b) Give an example of linearly independent functions that have linearly dependent sample vectors.
(c) Use this method to prove that the functions 1, cos x. sinx, cos2.x, sin2x, are linearly independent.