Question

Determine whether the functions y 1 and y 2 are linearly dependent on the interval​ (0,1). y 1equals2 cosine squared t minus 1​, y 2equals6 cosine 2 t Select the correct choice below​ and, if​ necessary, fill in the answer box within your choice. A. Since y 1equals(nothing )y 2 on​ (0,1), the functions are linearly independent on​ (0,1). ​(Simplify your​ answer.) B. Since y 1equals(nothing )y 2 on​ (0,1), the functions are linearly dependent on​ (0,1). ​(Simplify your​ answer.) C. Since y 1 is not a constant multiple of y 2 on​ (0,1), the functions are linearly dependent on​ (0,1). D. Since y 1 is not a constant multiple of y 2 on​ (0,1), the functions are linearly independent on​ (0,1).

Answer 1

D. Since y₁ is not a constant multiple of y₂ on​ (0,1), the functions are linearly independent on​ (0,1).

Explanation:

y₁=2Cos²t-1

y₂=6Cos2t

We want to determine if y₁ and y₂ are linearly independent in the interval (0,1).

To do this, we show that there does not exist any c₁ and c₂ in (0,1) that makes the expression:

f(t)=c₁(2Cos²t-1)+c₂6Cos2t=0.

If c₁ and c₂=0

f(t)=0

Let c₁=1 and c₂=-⅓

f(t)=c₁(2Cos²t-1)+c₂6Cos2t

=2Cos²t-1-⅓*6(cos²t-sin²t)

=2Cos²t-1-2cos²t+2sin²t

=2sin²t-1

Since f(t)≠0, y₁ is not a constant multiple of y₂ and the functions are linearly independent on (0,1).