10.1 are linear independent ....

[karush]

show that
$e^{2x},\quad \sin 2x$
are linear independent on
$\left\{-\infty,\infty\right\}$

new concept to me
but
$\sin 2x$
has an amplitude
$e^{2x}$
doesn't

 

[HOI]

Whether or not a "vector" has an "amplitude" is not relevant. All we require in a vector space is that we be able to add vectors and multiply vectors by numbers (more generally, "scalars").

Two vectors, u and v, in a vector space are "independent" if the only values or a and b such that au+ bv= 0 are a= b= 0. Here that gives us the equation $$ae^{2x}+ bsin(2x)= 0$$ for all x. In particular, if x= 0 that becomes $$a(1)+ b(0)= a= 0$$ and, if $$x= \pi/2$$, $$ae^{\pi/2}+ bsin(\pi/2)= 0$$ which, since a= 0, gives $$b= 0$$.

 

[karush]

 

[HOI]

?? That appears to be impossible to read!

 

[karush]

.