Practice Generalization To Function Spaces (23.12)

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8 questions

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6 sample questions

Question 1.
Identify if the functions 1, x, x² are linearly independent.
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Consider their definitions and how they relate.
Question 2.
Give an example of two linearly dependent functions.
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Look for functions that can create each other.
3Question 3.
Which of the following sets of functions is linearly independent?
- {1
- x}
- {x
- 2x}
- {2x
- x²}
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Think about relations between the functions.
4Question 4.
True or False: All polynomial functions are linearly independent.
- True
- False
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Check relationships between polynomials.
Question 5.
Given the functions f1(x) = e^x, f2(x) = e^(2x), and f3(x) = sin(x), determine if they are linearly independent.
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Use the Wronskian to check their independence.
Question 6.
In structural dynamics, if a certain mode shape is found to be linearly dependent on others in a system of vibrations, what potential issues could arise?
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Consider the implications of redundancy in engineering designs.

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