(C) Only 4. Let f1, f2, f3 be functions in F(R).a. For a set of real numbers x, X2, X3, let (f(x,)) be the 3-by-3 matrixwhose (i, j) entry is f(x;), for 1 < i, js 3. Prove that the functionsf1, f2, f3 are linearly independent if the rows of the matrix (f(x;)) arelinearly independent.b. Assume the functions f1, f2, f3 have first and second derivatives on someinterval (a, b), and let W(x) be the 3-by-3 matrix whose (i, j) entry isfU-), for 1 < i, j< 3, where f(0) = f, f(1) = f' and f(2) = f" for a dif-ferentiable function f. Prove that f, f2, f3 are linearly independent iffor some x in (a, b), the rows of the matrix W(x) are linearly independent.Show that the following sets of functions are linearly independent.c. f;(x) = -x² + x + 1, f2(x) = x² + 2x, f3(x) = x² – 1.d. f,(x) = e¯*, f2(x) = x, f,(x) = e2ª.e. fi(x) = e*, f2(x) = sin x, f3(x) = cos x.Note that if the tests in (a) or (b) fail, it is not guaranteed that the functions arelinearly dependent.%3D

Name: (C) Only 4. Let f1, f2, f3 be functions in F(R).a. For a set of real n
Uploaded: 2024-03-20T06:47:08.000Z
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Description: (C) Only 4. Let f1, f2, f3 be functions in F(R).a. For a set of real numbers x, X2, X3, let (f(x,)) be the 3-by-3 matrixwhose (i, j) entry is f(x;), for 1 < i, js 3. Prove that the functionsf1, f2, f3 are linearly independent if the rows of the matri