Linear independence of functions

Apr 2017
76
0
India
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?
 

topsquark

Math Team
May 2013
2,385
996
The Astral plane
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
 
  • Like
Reactions: 1 person
Dec 2015
937
122
Earth
Since the cos and sin are two different trigonometric functions then it proves it but the general method requires Wronskian calculation .