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4. Let f1, f2, f3 be functions in F(R). a. For a set of real numbers x1, X2, X3, let (f(x;)) be the 3-by-3 matrix whose (i, j) entry is f(x), for 1 < i, js 3. Prove that the functions f, f2, f3 are linearly independent if the rows of the matrix (f;(x;)) are linearly independent. b. Assume the functions f1, f2, f3 have first and second derivatives on some interval (a, b), and let W(x) be the 3-by-3 matrix whose (i, j) entry is SY-", 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 if for 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, f,(x) = x² – 1. d. f,(x) = e¯*, f2(x) = x, f,(x) = e²ª. e. f,(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 are linearly dependent. %3D

4. Let f1, f2, f3 be functions in F(R). a. For a set of real numbers x1, X2, X3, let (f(x;)) be the 3-by-3 matrix whose (i, j) entry is f(x), for 1 < i, js 3. Prove that the functions f, f2, f3 are linearly independent if the rows of the matrix (f;(x;)) are linearly independent. b. Assume the functions f1, f2, f3 have first and second derivatives on some interval (a, b), and let W(x) be the 3-by-3 matrix whose (i, j) entry is SY-", 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 if for 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, f,(x) = x² – 1. d. f,(x) = e¯*, f2(x) = x, f,(x) = e²ª. e. f,(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 are linearly dependent. %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 matrix whose (i, j) entry is f(x;), for 1 < i, js 3. Prove that the functions f1, f2, f3 are linearly independent if the rows of the matrix (f(x;)) are linearly independent. b. Assume the functions f1, f2, f3 have first and second derivatives on some interval (a, b), and let W(x) be the 3-by-3 matrix whose (i, j) entry is fU-), 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 if for 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 are linearly dependent. %3D
Transcribed Image Text: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 matrix whose (i, j) entry is f(x;), for 1 < i, js 3. Prove that the functions f1, f2, f3 are linearly independent if the rows of the matrix (f(x;)) are linearly independent. b. Assume the functions f1, f2, f3 have first and second derivatives on some interval (a, b), and let W(x) be the 3-by-3 matrix whose (i, j) entry is fU-), 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 if for 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 are linearly dependent. %3D
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