## Basis among trigonometric functions

In the space of real functions consider a subspace generated by functions $$\sin^2(x)$$, $$\sin(2x)$$, $$\cos^2(x), \cos(2x)$$ and $$f(x) = 1$$. Find a basis of this subspace.

• #### Resolution

Functions $$\cos(2x) = \cos^2(x) - \sin^2(x)$$ and $$\cos^2(x) = 1 - \sin^2(x)$$ are linearly dependent on functions $$M = \{f(x), \sin(2x), \sin^2(x)\}$$.

We show that $$M$$ is linearly independent. If $$M$$ was linearly dependent, then there would exist nontrivial $$a,b,c \in \mathbb R$$ such that for all $$x \in \mathbb R$$ holds

$$a f(x) + b \sin(2x) + c \sin^2(x) = 0 \hspace{1cm} (1).$$

We show that there is no nnontrivial $$a,b,c \in \mathbb R$$ that (1) would hold for $$x \in \{0, \pi/4, \pi/2\}$$.

Since the matrix

$$\begin{array}{c|ccc} x & 0 & \pi/4 & \pi/2 \\\hline f(x)=1 & 1 & 1 & 1 \\ \sin(2x) & 0 & 1 & 0 \\ \sin^2(x) & 0 & 1/2 & 1 \end{array}$$

is regular, according to the Frobenius theorem the system of three equations formed by subtituting $$x \in \{0, \pi/4, \pi/2\}$$ into (1) has only trivial solution $$a,b,c=0$$.

• #### Result

A basis could be e.g. chosen as $$f(x), \sin(2x)$$ and $$\sin^2(x)$$.