Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decision. {x°, x° - 1, 9} on (– 0,00) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The functions are linearly dependent because c,x° + c, (x° - 1) + C3 (9) = 0 has the solution c, = C2 and c, = - 1, and c3 = (Type integers or simplified fractions.) B. The functions are linearly independent because c, x° + c, (x° - 1) + ca (9) = 0 has no solutions for constants c,, c, C4X and C3 that are not all zero.
Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decision. {x°, x° - 1, 9} on (– 0,00) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The functions are linearly dependent because c,x° + c, (x° - 1) + C3 (9) = 0 has the solution c, = C2 and c, = - 1, and c3 = (Type integers or simplified fractions.) B. The functions are linearly independent because c, x° + c, (x° - 1) + ca (9) = 0 has no solutions for constants c,, c, C4X and C3 that are not all zero.
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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![**Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decision.**
Functions: \(\{x^5, x^5 - 1, g\}\) on \((- \infty, \infty)\).
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Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- **A.** The functions are linearly dependent because \( c_1 x^5 + c_2 \left(x^5 - 1\right) + c_3 (9) = 0 \) has the solution \( c_1 = \) [box] and \( c_2 = -1 \), and \( c_3 = \) [box].
*(Type integers or simplified fractions.)*
- **B.** The functions are linearly independent because \( c_1 x^5 + c_2 \left(x^5 - 1\right) + c_3 (9) = 0 \) has no solutions for constants \( c_1, c_2, \) and \( c_3 \) that are not all zero.
*(Select and provide justification based on calculations or known theoretical results.)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4c9d77e-368b-42fe-b276-8a66252e95fb%2Fe5bc71c2-bb4d-4bbb-b904-19f84caf7e2f%2Fc9x39u_processed.png&w=3840&q=75)
Transcribed Image Text:**Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decision.**
Functions: \(\{x^5, x^5 - 1, g\}\) on \((- \infty, \infty)\).
---
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- **A.** The functions are linearly dependent because \( c_1 x^5 + c_2 \left(x^5 - 1\right) + c_3 (9) = 0 \) has the solution \( c_1 = \) [box] and \( c_2 = -1 \), and \( c_3 = \) [box].
*(Type integers or simplified fractions.)*
- **B.** The functions are linearly independent because \( c_1 x^5 + c_2 \left(x^5 - 1\right) + c_3 (9) = 0 \) has no solutions for constants \( c_1, c_2, \) and \( c_3 \) that are not all zero.
*(Select and provide justification based on calculations or known theoretical results.)*
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