Which of the following statements are true?

Statement A: The set of functions
{
𝑒
𝑡
,
𝑒
2
𝑡
}
is linearly independent on
𝑅
.

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

Statement C: The set of functions
{
1
,
𝑡
2
}
is linearly independent on
𝑅
.

Question

Which of the following statements are true?

Statement A:  The set of functions  
{
𝑒
𝑡
,
𝑒
2
𝑡
}
  is linearly independent on  
𝑅
.

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

Statement C:  The set of functions  
{
1
,
𝑡
2
}
  is linearly independent on  
𝑅
.

BlackTom AI · Accepted Answer

Consider the question about which statements are true.
Option A discusses the set {e^t, e^{2t}} on R. To test independence, suppose c1 e^t + c2 e^{2t} = 0 for all t. This is a linear combination of two distinct exponential functions with different growth rates. Evaluating at different t-values or using the fact that e^t and e^{2t} are not scalar multiples of each other shows that the only solution is c1 = c2 = 0, so the functions are linearly independent. Therefore, Statement A is true.

Option B concerns {sin(3t), cos(3t)} on the interval (0, 2π). A standard result in real analysis is that sine and cosine of the same frequency are linearly independent as functions: if a·sin(3t) + b·cos(3t) = 0 for all t in the interval, then a = 0 and b = 0. Hence the pair is linearly independent, not dependent. This makes Statement B false.

Option C looks at {1, t^2} on R. If a + b t^2 = 0 for all t, then evaluating at t = 0 gives a = 0, and comparing coefficients shows b must also be 0. Therefore the only solution is a = b = 0, and the set is linearly independent. Thus Statement C is true.

In summary, A is true because e^t and e^{2t} are not scalar multiples; B is false because sin(3t) and cos(3t) are independent; C is true because 1 and t^2 are not constant multiples of each other over R.

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