In the vector space of all real-valued functions, find a
basis for the subspace spanned by {sin(t); sin(2t); sin(t)cos(t)}
I can clearly see that sin(2t) can be written as 2*sin(t)cos(t), so that means that 2*sin(t)cos(t) and sin(t)cos(t) are linearly dependent sets.
Am I right so far?
So where do I go from here?
I want to add to topsquarks post.
Since he does not explain why,
{sin(t),sin(t)cos(t)}
Those two a linearly independent (and hence for a basis for its subspace).
It is because these are not konstant multiples of each other.
Since,
sin(t)!=ksin(t)cos(t)
Using the special theorem for exactly two vectors.