23.12. Generalization to Function Spaces

Try these first

1. 1.

   Identify if the functions 1, x, x² are linearly independent.

   Hint

   Consider their definitions and how they relate.

2. 2.

   Give an example of two linearly dependent functions.

   Hint

   Look for functions that can create each other.

3. 3.

   Which of the following sets of functions is linearly independent?

   • {1
   • x}
   • {x
   • 2x}
   • {2x
   • x²}

   Hint

   Think about relations between the functions.

4. 4.

   True or False: All polynomial functions are linearly independent.

   • True
   • False

   Hint

   Check relationships between polynomials.

5. 5.

   Given the functions f1(x) = e^x, f2(x) = e^(2x), and f3(x) = sin(x), determine if they are linearly independent.

   Hint

   Use the Wronskian to check their independence.

6. 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?

   Hint

   Consider the implications of redundancy in engineering designs.