Let $C[-\pi, \pi]$ be the vector space of all continuous functions defined on the interval $[-\pi, \pi]$.

Show that the subset $\{\cos(x), \sin(x)\}$ in $C[-\pi, \pi]$ is linearly independent.

Proof.

Note that the zero vector in the vector space $C[-\pi, \pi]$ is the zero function
\[\theta(x):=0.\]

Let us consider a linear combination
\[a_1\cos(x)+a_2\sin(x)=\theta(x)=0 \tag{*}.\] If this linear combination has only the zero solution $a_1=a_2=0$, then the set $\{\cos(x), \sin(x)\}$ is linearly independent.

The equality (*) should be true for any values of $x\in [-\pi, \pi]$.
Setting $x=0$, we obtain from (*) that
\[a_1=0\] since $\cos(0)=1, \sin(0)=0$.

We also set $x=\pi/2$ and we obtain
\[a_2=0\] since $\cos(\pi/2)=0, \sin(\pi/2)=1$.

Therefore, we have $a_1=a_2=0$ and we conclude that the set $\{\cos(x), \sin(x)\}$ is linearly independent.

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