10.1 are linear independent ....

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Discussion Overview

The discussion revolves around the linear independence of the functions $e^{2x}$ and $\sin 2x$ over the interval $\left\{-\infty,\infty\right\}$. Participants explore the concept of linear independence in the context of vector spaces and provide reasoning related to the properties of these functions.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant suggests that the presence of an amplitude in $\sin 2x$ and its absence in $e^{2x}$ indicates a difference that may relate to their linear independence.
  • Another participant argues that the concept of amplitude is not relevant to determining linear independence, emphasizing that linear independence is defined by the ability to express a linear combination of vectors equaling zero only when all coefficients are zero.
  • The second participant provides a mathematical approach to demonstrate linear independence by evaluating the equation $ae^{2x} + b\sin(2x) = 0$ at specific values of $x$.
  • A later reply expresses confusion regarding the clarity of the previous post, indicating a potential communication issue.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus. There are differing views on the relevance of amplitude to linear independence, and the clarity of the mathematical explanation is questioned.

Contextual Notes

Some assumptions regarding the definitions of linear independence and vector spaces may not be fully articulated, and the discussion includes varying levels of mathematical clarity.

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
 
Last edited:
?? That appears to be impossible to read!
 
.
 
Last edited:

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