Basis among trigonometric functions
Task number: 2548
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 \(\sin^2(x)\).
