Question

Determine if the following sets are linearly dependent, or independent.

(i) {1,sin(x),cos(x)}

(ii) {sin²(x),cos(2x),cos²(x)}

Answer 1

Let us determine if the following sets are linearly dependent, or independent.

(i) $\{1,\sin(x),\cos(x)\}$

Since the Wronskian

$W(1,\sin(x),\cos(x))=\begin{vmatrix} 1 & \sin(x) & \cos(x)\\ 0 & \cos(x) & -\sin(x)\\ 0 & -\sin(x) & -\cos(x) \end{vmatrix} = \begin{vmatrix} \cos(x) & -\sin(x)\\ -\sin(x) & -\cos(x) \end{vmatrix} \\=-\cos^2(x)-\sin^2(x)=-1\ne 0,$ ∣

we conclude that the set $\{1,\sin(x),\cos(x)\}$ is linearly independent.

(ii) $\{\sin^2(x),\cos(2x),\cos^2(x)\}$

Taking into account that $\cos(2x)=1\cdot\cos^2(x)-1\cdot\sin^2(x),$ we conclude that this set is linearly dependent.