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May 15th, 2019, 09:52 AM   #1
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Linear independence of functions

Whether Cos 3x and Cos (3x + Pi/2)are linearly independent or not in the interval (-infinity, infinity)?

My attempt:

I calculated the Wronskian that comes out to be -3 (independent of x) that signifies that this function is linearly independent because the sufficient condition for the set of functions to be linearly independent is the non-vanishing of Wronskian for atleast one point in the interval.

Am I correct?
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May 15th, 2019, 10:45 AM   #2
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Originally Posted by shashank dwivedi View Post
Whether Cos 3x and Cos (3x + Pi/2)are linearly independent or not in the interval (-infinity, infinity)?

My attempt:

I calculated the Wronskian that comes out to be -3 (independent of x) that signifies that this function is linearly independent because the sufficient condition for the set of functions to be linearly independent is the non-vanishing of Wronskian for atleast one point in the interval.

Am I correct?
Yes. Also $\displaystyle cos \left ( 3x + \dfrac{\pi}{2} \right ) = sin(3x)$ which makes this a bit easier on the calculation. (At least in my own mind.)

-Dan
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May 15th, 2019, 01:32 PM   #3
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Since the cos and sin are two different trigonometric functions then it proves it but the general method requires Wronskian calculation .
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