Linear Independence: Proving SIN(nx) is Indep.

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SUMMARY

The discussion centers on proving the linear independence of the set of functions SIN(nx) for n = 1, 2, 3, etc. It is established that no linear combination of the functions SIN(n1 x) and SIN(n2 x) can yield SIN(n3 x) when n1, n2, and n3 are distinct integers. This confirms that the functions are linearly independent, as the only solution to the equation involving these functions is the trivial solution where all coefficients are zero.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically linear independence.
  • Familiarity with trigonometric functions, particularly sine functions.
  • Basic knowledge of Fourier series and their properties.
  • Ability to manipulate and solve equations involving functions.
NEXT STEPS
  • Study the properties of linear independence in vector spaces.
  • Explore the implications of linear independence in Fourier series.
  • Learn about orthogonal functions and their role in function spaces.
  • Investigate the application of trigonometric identities in proving independence.
USEFUL FOR

Mathematicians, physics students, and anyone studying linear algebra or Fourier analysis will benefit from this discussion.

mkbh_10
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Show that the set of functions SIN(nx) where n=1,2,3... is linearly independent ?
 
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Can you add sin(n1 x) and sin(n2 x) to get sin(n3 x)
 
nope , therefore linear independnce
 

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