MHB 10.1 are linear independent ....

karush
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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
 
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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.
 
SSCwt.png
 
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?? That appears to be impossible to read!
 
.
 
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