Final answer:
S₁ is True as the presence of the zero function ensures that the set is linearly dependent. S₂ is False and should be corrected to state that a set of functions is linearly independent if and only if no function in the set can be written as a linear combination of the others.
Step-by-step explanation:
Let's evaluate the statements S₁ and S₂:
- S₁: The set of functions containing the zero function is Linearly Dependent.
- S₂: The set of functions containing linearly independent functions is Linearly Independent.
A set of functions (or vectors) is considered linearly dependent if at least one of the functions (or vectors) can be expressed as a linear combination of the others. Since the zero function can be represented as a linear combination of any functions (simply by multiplying them by zero), the presence of the zero function makes the set linearly dependent. Therefore, S₁ is True.
Statement S₂ can be a bit tricky. If the set contains functions that are each linearly independent on their own, this does not automatically guarantee that the set as a whole is linearly independent. A set of functions is linearly independent if no function in the set can be written as a linear combination of the others. The statement in S₂ omits the important detail that the functions must be linearly independent from each other. Thus, without this clarification, S₂ is False. The correct statement would be: 'The set of functions is linearly independent if and only if no function in the set can be written as a linear combination of the others.'