Question determine which of the sets of vectors are linearly

Question

determine which of the sets of vectors are linearly independent
A: the set {sin t , tan t} in C[0,1]
B: The set {sin(t) cos(t) , cos (2t)} C[0,1]
C: the set {cos^2(t), 1+cos (2t)} c[0,1]

Solution

Rule: {a,b} is a set of containing 2 vectors and they will be linearly independent if one element is not a scalar multiple of other.

A) tan t = sin t/cos t = sect sin t , so tan t can be expressed as a multiple of sin t , so { sin t, tan t} is not linearly independent.

C) cos 2t =2 cos^2 (t) -1 => 1+cos 2t = 2 cos^2 (t)

so 1+ cos 2t is a scalar multiple of cos^2 (t) . Thus it\'s not linearly independent.

B)

sint cost = 1/2 (2 sin t cos t) =1/2 sin 2t

As 1/2 sin2t = sin t cos t and cos 2t are not scalar multiple to one another, the set is linearly independent.