(a.) Explanation Given: A set of functions S and an interval I . The following statements: If S contains the zero function, then S can be linearly independent. If S is linearly independent on a subinterval J of I , then it is linearly independent on I . If S is linearly dependent on a subinterval J of I , then it is linearly dependent on I . If S is linearly independent on I , then it is linearly independent on a subinterval J of I . If S is linearly dependent on I , then it is linearly dependent on a subinterval J of I . If S is linearly dependent on I , and if T contains S , then T is linearly dependent on I . Concept used: A set of functions, S = f 1 x , f 2 x , ... , f n x is linearly independent on an interval I if and only if c 1 f 1 x + c 2 f 2 x + ... + c n f n x = 0 for all x ∈ I , implies that all scalars c i are zero. The set of functions is linearly dependent if it is not linearly independent. Calculation: The first statement is given as follows: “If S contains the zero function, then S can be linearly independent.” Let S = f 1 x , f 2 x , ... , f n x contain the zero function. It can be assumed arbitrarily that f k = 0 . Now, let c i = 0 for all i ≠ k . That is, c k ≠ 0 . Then, clearly c 1 f 1 x + c 2 f 2 x + ... + c k f k x + ... + c n f n x = 0 , despite all scalars not being zero ( c k ≠ 0 ). This violates the condition for linear independence. So, if S contains the zero function, then S must be linearly dependent. Thus, the first statement is not true. The second statement is given as follows: “If S is linearly independent on a subinterval J of I , then it is linearly independent on I .” Let S = f 1 x , f 2 x , ... , f n x be linearly independent on a subinterval J of I . Then, by definition, c 1 f 1 x + c 2 f 2 x + ... + c n f n x = 0 for all x ∈ J , implies that all scalars c i are zero. Now, let c 1 f 1 x + c 2 f 2 x + ... + c n f n x = 0 for all x ∈ I . Then, obviously c 1 f 1 x + c 2 f 2 x + ... + c n f n x = 0 for all x ∈ J ⊂ I . Now, as determined previously, it follows that all scalars c i are zero. So, c 1 f 1 x + c 2 f 2 x + ... + c n f n x = 0 for all x ∈ I , implies that all scalars c i are zero. Then, if S is linearly independent on a subinterval J of I , then it is linearly independent on I . Thus, the second statement is true. The third statement is given as follows: “If S is linearly dependent on a subinterval J of I , then it is linearly dependent on I .” Let S = f x , g x , where f x = 1 0 < x < 2 0 otherwise and g x = x 1 < x < 3 0 otherwise . Let J = 0 , 1 and I = 0 , 3 . Clearly, J is a subinterval of I . Now, according to the function definitions, when x ∈ J = 0 , 1 , f x = 1 and g x = 0 . As determined previously, any set of functions containing the zero function, is linearly dependent. So, S = f x , g x is linearly dependent on a subinterval J = 0 , 1 of I = 0 , 3 . Similarly, according to the function definitions, when x ∈ I = 0 , 3 , more specifically, when x ∈ 1 , 2 , f x = 1 and g x = x . Now, c 1 f x + c 2 g x = c 1 + c 2 x = 0 implies that c 1 = c 2 = 0 . So, c 1 f x + c 2 g x = 0 for all x ∈ 1 , 2 implies that c 1 = c 2 = 0 . Further, c 1 f x + c 2 g x = 0 for all x ∈ I = 0 , 3 implies that c 1 = c 2 = 0 . This implies that S is linearly independent on I . Then, if S is linearly dependent on a subinterval J of I , then it is not necessarily linearly dependent on I . Thus, the third statement is not true. The fourth statement is given as follows: “If S is linearly independent on I , then it is linearly independent on a subinterval J of I .” Let S = f x , g x , where f x = 1 0 < x < 2 0 otherwise and g x = x 1 < x < 3 0 otherwise (b.) To determine The cases in which the Wronskian can be used for testing linear independence and any other means by which such a test can be performed.

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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Question

Chapter 3.1, Problem 16P

(a.)

To determine

Whether the given statements are true; with proof and example.

(b.)

To determine

The cases in which the Wronskian can be used for testing linear independence and any other means by which such a test can be performed.

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Chapter 3.1, Problem 15P

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Chapter 3 Solutions

Advanced Engineering Mathematics

Ch. 3.1 - 1–6 BASES: TYPICAL EXAMPLES To get a feel for...Ch. 3.1 - 1–6 BASES: TYPICAL EXAMPLES To get a feel for...Ch. 3.1 - 1–6 BASES: TYPICAL EXAMPLES To get a feel for...Ch. 3.1 - 1–6 BASES: TYPICAL EXAMPLES To get a feel for...Ch. 3.1 - Prob. 5P Ch. 3.1 - Prob. 6P Ch. 3.1 - Prob. 8P Ch. 3.1 - Prob. 9P Ch. 3.1 - Prob. 10P Ch. 3.1 - Prob. 11P

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