Which of the following statements are true?

Statement A: The set of functions {et,e2t} is linearly independent on R.

Statement B: The set of functions {sin(3t),cos(3t)} is linearly dependent on (0,2π).

Statement C: The set of functions {1,t2} is linearly independent on R.

Question

Which of the following statements are true?

Statement A:  The set of functions  {et,e2t}  is linearly independent on  R.

Statement B:  The set of functions  {sin(3t),cos(3t)}  is linearly dependent on  (0,2π).

Statement C:  The set of functions  {1,t2}  is linearly independent on  R.

BlackTom AI · Accepted Answer

Let’s parse each statement about linear independence carefully and test the given sets.

Option A analyzes the pair {e^t, e^{2t}} on R. If a·e^t + b·e^{2t} = 0 for all t, then dividing by e^t (which is never zero) gives a + b·e^t = 0 for all t. Since e^t takes on a continuum of positive values, the only way this equality can hold for all t is a = 0 and b = 0. Therefore, the functions e^t and e^{2t} are linearly independent on R. This makes Statement A true.

Option B looks at the pair {sin(3t), cos(3t)} on (0, 2π). A quick check is to see if one is a scalar multiple of the other. Sin(3t) and cos(3t) are not scalar multiples of each other; moreover, the Wronskian W = sin(3t)·(−3sin(3t)) − cos(3t)·(3cos(3t)) = −3(sin^2(3t) + cos^2(3t)) = −3, which is nonzero for all t. A nonzero Wronskian on an interval implies linear independence of the functions on that interval. Hence, sin(3t) and cos(3t) are linearly independent on (0, 2π). Therefore, Statement B is false, since it claims dependence.

Option C considers the pair {1, t^2} on R. If a·1 + b·t^2 = 0 for all t, then this is a polynomial identity forcing a = 0 and b = 0, because the constant term must vanish independently of the coefficient of t^2, and vice versa. Thus 1 and t^2 are linearly independent on R. This makes Statement C true.

Summarizing the analysis of the individual statements: 
- A is true because e^t and e^{2t} are not scalar multiples and only the trivial combination yields zero for all t. 
- B is false because sin(3t) and cos(3t) are not multiples of each other and have a nonzero Wronskian on the interval, confirming independence. 
- C is true because 1 and t^2 cannot combine with nonzero coefficients to produce the zero function for all t.

Considering which statements are true, A and C are true while B is false. The provided answer option that lists the true statements accordingly is the one that identifies A and C as the true pair.

Which of the following statements are true? Statement A: The set of functions {et,e2t} is linearly independent on R. Statement B: The set of functions {sin(3t),cos(3t)} is linearly dependent on (0,2π). Statement C: The set of functions {1,t2} is linearly independent on R.单项选择题

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