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mkbh_10
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Show that the set of functions SIN(nx) where n=1,2,3... is linearly independent ?
Linear independence is a concept in linear algebra that describes the relationship between vectors in a vector space. It means that no vector in the set can be expressed as a linear combination of the other vectors in the set.
To prove linear independence, you need to show that there is no way to create a linear combination of the vectors that equals the zero vector, except by setting all the coefficients to zero. This can be done through various methods, such as Gaussian elimination or using the definition of linear independence.
SIN(nx) is a trigonometric function that represents a sine wave with a frequency of n. It is related to linear independence because it is a set of vectors in a vector space that can be used to prove linear independence.
To prove that SIN(nx) is linearly independent, you would need to show that there is no way to create a linear combination of the sine waves with different frequencies that equals the zero vector. This can be done by setting up a system of equations and showing that the only solution is when all the coefficients are equal to zero.
Proving that SIN(nx) is linearly independent is important because it provides a concrete example of linear independence in action. It also has practical applications, such as in signal processing and Fourier analysis, where understanding linear independence is crucial for accurate calculations and predictions.