Linear dependence in vector spaces. Which one is true? I am trying to determine whether $\{ \sin x, \cos x\}$ is linearly dependent in vector space of all real valued functions. The definition says:

A set of vectors $\{ \vec v_1, \dots, \vec v_k \}$ in a vector space $V$ is linearly dependent if there are scalars $c_1, \dots, c_k$, at least one of which is not zero, such that
$$c_1 \vec v_1 + \dots + c_k \vec v_k=\vec 0 $$

But considering the above example, does it mean that if $\{ \sin x, \cos x\}$ is linearly dependent, then
a) there exist scalars $c,d$ (at least one of which is not zero) such that $c\sin x+d\cos x=0$ for every $x$
or
b) for every $x$, there exist scalars $c,d$ (at least one of which is not zero) such that $c\sin x+d\cos x=0$
I can't tell this from the definition. Which one is true?
 A: Call $f$ the function $f(x) = \cos(x)$ and $g$ the function $g(x) = \sin(x)$. Then, you need to prove that
$$
c f + d g = 0 \Longrightarrow c = d = 0,
$$
where in the identity $cf + dg = 0$, $0$ is actually the origin of your vector space, i.e. the constant function $0$. In other words, $cf + d g = 0$ means
$$
c f(x) + d g(x) = 0, \quad \forall x \in \mathbb{R},
$$
hence you need to prove that
$$
[c \cos(x) + d \sin(x) = 0, \quad \forall x \in \mathbb{R}] \Longrightarrow c = d = 0.
$$
To prove this, just consider $x = 0$ and $x = \pi/2$.
A: $\sin x$ and $\cos x$ are real numbers when $x$ is real. The vectors in the vector space of real valued functions you are talking about are $\sin$ and $\cos$. These both vectors are linearly dependants iff by definition there are two real $a,b$ with at least one which is not zero such that
$$a\sin +b\cos =0$$
and the above equality holds in the vector space of the real valued functions (it is the identically zero function). It means that for all $x\in\mathbb{R}$, we have
$$a\sin x +b\cos x=0$$
and this last equality holds in $\mathbb{R}$.
