Can i say that translation and / or rotation of a curve is linearly dependent of initial curve?
I was thinking of problem $\{ \sin x, \cos x\}$ are linearly dependent or not? I found both are linearly dependent as constants are zero. Here $\cos x$ is just $\sin x$ curve shifted to right by 90 degrees right. It is just translation.
Then $\{x , x+1 \}$ both are linearly dependent.
Can i say that if i translate and or rotate a curve, both are linearly dependent? Is this generalized statement valid?
If i take another example, $\{ \sin x, \cos x, \tan x\}$, i am getting constants as zero which means linearly independent. But on other side, $\tan x = \frac {\sin x}{ \ cos x}$
Say in set of vectors {u,v,w}, if w = u/v, what can be said of linear dependence or independence?