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Which of the following sets of functions are linearly independent on (0, o)? Select all that apply. O {7, sin? x, cos² x} O {1, x + 3, 2x} O {vx, x, x2} O {2, tan? x, sec2 x}

Which of the following sets of functions are linearly independent on (0, o)? Select all that apply. O {7, sin? x, cos² x} O {1, x + 3, 2x} O {vx, x, x2} O {2, tan? x, sec2 x}

Advanced Engineering Mathematics
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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**Question:** Which of the following sets of functions are linearly independent on \( (0, \, \infty) \)? Select all that apply. 1. \(\{7, \sin^2 x, \cos^2 x\}\) 2. \(\{1, x + 3, 2x\}\) 3. \(\{\sqrt{x}, x, x^2\}\) 4. \(\{2, \tan^2 x, \sec^2 x\}\) **Explanation:** Consider each set of functions for linear independence over the interval \( (0, \, \infty) \): 1. **\(\{7, \sin^2 x, \cos^2 x\}\):** - Note that \(\sin^2 x + \cos^2 x = 1\), which implies these are not linearly independent. 2. **\(\{1, x + 3, 2x\}\):** - Functions like \(1\) and \(2x\) are clearly distinct and cannot be represented as linear combinations of one another. Assume \(\{1, x + 3, 2x\}\) are linearly independent. 3. **\(\{\sqrt{x}, x, x^2\}\):** - Consider the polynomial nature and function powers. All functions are constructed from distinct powers of \(x\), suggesting linear independence. 4. **\(\{2, \tan^2 x, \sec^2 x\}\):** - Recall \(\sec^2 x = \tan^2 x + 1\). This relationship implies linear dependence.
Transcribed Image Text:**Question:** Which of the following sets of functions are linearly independent on \( (0, \, \infty) \)? Select all that apply. 1. \(\{7, \sin^2 x, \cos^2 x\}\) 2. \(\{1, x + 3, 2x\}\) 3. \(\{\sqrt{x}, x, x^2\}\) 4. \(\{2, \tan^2 x, \sec^2 x\}\) **Explanation:** Consider each set of functions for linear independence over the interval \( (0, \, \infty) \): 1. **\(\{7, \sin^2 x, \cos^2 x\}\):** - Note that \(\sin^2 x + \cos^2 x = 1\), which implies these are not linearly independent. 2. **\(\{1, x + 3, 2x\}\):** - Functions like \(1\) and \(2x\) are clearly distinct and cannot be represented as linear combinations of one another. Assume \(\{1, x + 3, 2x\}\) are linearly independent. 3. **\(\{\sqrt{x}, x, x^2\}\):** - Consider the polynomial nature and function powers. All functions are constructed from distinct powers of \(x\), suggesting linear independence. 4. **\(\{2, \tan^2 x, \sec^2 x\}\):** - Recall \(\sec^2 x = \tan^2 x + 1\). This relationship implies linear dependence.
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