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Are the function $f_1(x)=1,f_2(x)=\sin x,f_3(x)=\sin^2x$ linearly independent?

My try:I calculated the Wronskian and got $2\cos x\cos 2x+\sin x\sin 2x$.so what should I conclude and why?

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  • $\begingroup$ What is the space? $\endgroup$ – Olba12 Mar 9 '17 at 7:14
  • $\begingroup$ Nothing is mentioned. $\endgroup$ – MatheMagic Mar 9 '17 at 7:15
  • $\begingroup$ And the Field is $\mathbb{R}$? $\endgroup$ – Olba12 Mar 9 '17 at 7:17
  • $\begingroup$ yes ,i think so. $\endgroup$ – MatheMagic Mar 9 '17 at 7:18
  • $\begingroup$ Is the definition of linear independence different than for finite vector spaces? $\endgroup$ – mvw Mar 9 '17 at 7:22
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Assume that $c_1.f_1+c_2.f_2+c_3.f_3=0\implies c_1.1+c_2\sin x+c_3.\sin^2x=0$.

Putting $x=0;\implies c_1=0$.

Putting $x=\frac{\pi}{2}\implies c_2+c_3=0$;

Putting $x=-\frac{\pi}{2}\implies -c_2+c_3=0$;

Solving the above two equations give $c_2=c_3=0$

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    $\begingroup$ you mean to say linearly independent.Right? $\endgroup$ – MatheMagic Mar 9 '17 at 7:28
  • $\begingroup$ Yes that's right@MatheMagic $\endgroup$ – Learnmore Mar 9 '17 at 7:28

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