Explanation Given: The given set of functions is: F = { x 2 , x | x | } . Approach: The Wronskian of set of functions F is given by, W ( x ) = | f 0 ( x ) f 1 ( x ) f 0 ( 1 ) ( x ) f 1 ( 1 ) ( x ) | The set F is linearly independent in [ a , b ] if and only if its Wronskian is never zero for any f in [ a , b ] . If the Wronskian W ( x ) = 0 , then use the definition of the linearly independence. The set of functions is linearly independent if there exist scalars a 1 , a 2 , a 3 , ... , a n all are zero such that a 1 f 1 ( x ) + a 2 f 2 ( x ) + ... + a n f n ( x ) = 0 . Calculation: Here, the set of functions F is { x 2 , x | x | } . Thus, f 0 ( x ) = x 2 f 1 ( x ) = x | x | Split the given interval in [ − 1 , 0 ) and [ 0 , 1 ] . Evaluate the Wronskian of F in the interval [ − 1 , 0 ) as follows, W ( x ) = | x 2 x ( − x ) 2 x − 2 x | = x 2 ( − 2 x ) + 2 x ( x 2 ) = − 2 x 3 + 2 x 3 = 0 Evaluate the Wronskian of F in the interval [ 0 , 1 ] as follows,

Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

5th Edition

ISBN: 9780134689531

Author: Lee Johnson, Dean Riess, Jimmy Arnold

Publisher: PEARSON

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Chapter 6.5, Problem 15E

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The Wronskian of {x2,x|x|} and also determine whether the set of is linearly independent on the interval [−1,1] or not.

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Chapter 6.5, Problem 14E

Chapter 6.5, Problem 16E

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Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

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The Wronskian of { x 2 , x | x | } and also determine whether the set of is linearly independent on the interval [ − 1 , 1 ] or not.

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To calculate: The Wronskian of {x2,x|x|} and also determine whether the set of is linearly independent on the interval [−1,1] or not.

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To calculate:

The Wronskian of {x2,x|x|} and also determine whether the set of is linearly independent on the interval [−1,1] or not.