# Is the set $\{f_1,f_2,f_3\}$ linearly independent?

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?

• What is the space? – Olba12 Mar 9 '17 at 7:14
• Nothing is mentioned. – MatheMagic Mar 9 '17 at 7:15
• And the Field is $\mathbb{R}$? – Olba12 Mar 9 '17 at 7:17
• yes ,i think so. – MatheMagic Mar 9 '17 at 7:18
• Is the definition of linear independence different than for finite vector spaces? – mvw Mar 9 '17 at 7:22

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 $c_2=c_3=0$