Explanation Given: The set the functions is, { 1 , x , x 2 , ..... , x n } Proof: Consider the following set of functions: { 1 , x , x 2 , ..... , x n } Consider the polynomial in the interval ( a , b ) as, p ( x ) = a 0 + a 1 x + a 2 x 2 + ...... + a n x n Consider for all x ∈ ( a , b ) , p ( x ) = 0 Because for all values of x in the interval ( a , b ) , p ( x ) = 0 , thus all the values in ( a , b ) is the root of the equation p ( x ) = 0

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 6.1, Problem 27E

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The set of functions {1,x,x2,.....,xn}, where n is a positive integer is linearly independent on every open interval (a,b)

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Chapter 6.1, Problem 26E

Chapter 6.1, Problem 28E

Chapter 6 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - Prob. 7E Ch. 6.1 - In Problems7-14, determine whether the given...Ch. 6.1 - In Problems7-14, determine whether the given...Ch. 6.1 - In Problems7-14, determine whether the given...

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The set of functions { 1 , x , x 2 , ..... , x n } , where n is a positive integer is linearly independent on every open interval ( a , b )

Solution Summary: The author proves that n is a positive integer linearly independent on every open interval.

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To show:

The set of functions {1,x,x2,.....,xn}, where n is a positive integer is linearly independent on every open interval (a,b)

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Chapter 6 Solutions

The set of functions { 1 , x , x 2 , ..... , x n } , where n is a positive integer is linearly independent on every open interval ( a , b )

Solution Summary: The author proves that n is a positive integer linearly independent on every open interval.

Concept explainers

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To show: The set of functions {1,x,x2,.....,xn}, where n is a positive integer is linearly independent on every open interval (a,b)

Want to see the full answer?

Chapter 6 Solutions

To show:

The set of functions {1,x,x2,.....,xn}, where n is a positive integer is linearly independent on every open interval (a,b)