Find, if the following functions are linearly independent in vector space $F(\mathbb{R},\mathbb{R}) $: $f=Id, g(x) =\sin x , h(x)=\cos x. $

So I have $f:x \mapsto x, g:x \mapsto \sin x, h:x \mapsto \cos x$.

If functions are linearly independent we should receive $\alpha = \beta=\gamma =0$ from the following equation:

$$ \alpha \cdot \begin{bmatrix}x\\x\end{bmatrix} + \beta \cdot \begin{bmatrix}x\\ \sin x\end{bmatrix} + \gamma \cdot \begin{bmatrix}x\\ \cos x\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix} $$

How to solve it? We have 2 equations and we don't know $\alpha, \beta, \gamma$ or $x$?

  • $\begingroup$ What exactly is your definition of $F(\mathbb R,\mathbb R)$? $\endgroup$
    – Math1000
    Nov 28 '16 at 18:19
  • $\begingroup$ I don't know. I wrote what I had received from professor. $\endgroup$ Nov 28 '16 at 18:22

The trick is that this is a functional equation, meaning it has to hold for all values of $x \in \mathbb{R}$.

The second equation is enough. If you must have $$ ax + b\sin x + c \cos x = 0 $$ for all values of $x$, then at $x=0$ you have $c = 0$ and your equation reduces to $$ax + b\sin x =0.$$ At $x = \pi$, you imply $a=0$ and that in turn forces $b=0$.


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