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
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Linear Transformations
Section6.4: Transistion Matrices And Similarity
Problem 15E
Related questions
Topic Video
Question
100%
(d) only
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,