There is a lot of posts about that subject, but I'm not sure.
Task: Check if $$f(x)=\sin(x) $$ and $$g(x)=\cos(x) $$
are linearly independent in the space of functions.
That's what I have done:
$a \sin(x) + b \cos(x) = 0$, so $a=0$ and $b=0$.
for $x=0 $, $a \sin(0) + b \cos(0) = 0$, then $a\cdot 0 + b\cdot 1 = 0$.
for $x=\frac{\pi}{2}$
$a \sin(\frac{\pi}{2}) + b \cos(\frac{\pi}{2}) = 0$, then $b \cdot 1 + a \cdot 0 = 0$
what am I supposed to write then? :)
On
The Wronskian of $\sin$ and $\cos$ is not zero:
$$W(\sin, \cos)(x) = \begin{vmatrix} \sin x & \cos x \\ \cos x & -\sin x\end{vmatrix} = -\sin^2 x - \cos^2 x = -1 \ne 0$$
Therefore $\{\sin, \cos\}$ is linearly independent.
You have the right ideas but you need to write your solution more clearly. We are trying to show that if for all $x$ $$ a\sin(x) + b\cos(x) = 0\tag{1} $$ then $a=0$ and $b=0$. In particular (1) holds for $x=0$ and $x=\pi/2$. Substituting these into (1) we get $$ \begin{align} a(0)+b(1)&=0\\ a(1)+b(0)&=0. \end{align} $$ The first equation says that $b=0$ and the second that $a=0$.