How to know if a function is linearly independent or dependent? Original Problem: Determine if the set of functions 
$$\{ y_1(x),y_2(x),y_3(x) \} = \{x^2, \sin x, \cos x \}$$
 is linearly independent.
I understand I have to use the Wronskian method, but how would it work for three functions with sine and cosine? Can someone help me give a brief overview of what I need to do and does the terms actually cancel?
$$W(y_1,y_2,y_3)(x) = \det \begin{pmatrix}
x^2 & \sin x & \cos x \\
2x & \cos x & -\sin x \\
2 & -\sin x & -\cos x 
\end{pmatrix}$$
 A: I think you can use a more elementary method.
$\{x^2,\sin x, \cos x\}$ are linearly independent iff $ax^2+b\sin x+c \cos x=0 \implies a,b,c=0$.
But, if $ax^2+b\sin x+c \cos x=0$, then, since $x^2$ is unlimited while $\sin x$ and $\cos x$ are not, $a=0$.
Also $\sin (0)=0, \cos(0)=1 \implies c=0$. So, $a,b,c=0$. 
A: $ax^2+b\cos x+c\sin x=0\,,\forall x\implies a(0)^2+b\cos0+c\sin0=0\implies b=0$.
So we have $ax^2+c\sin x=0\,,\forall x\implies a( \pi)^2+c\sin (\pi)=0\implies a(\pi)^2=0\implies a=0$.
So we have $c\sin x=0\,,\forall x\implies c=0$.
A: $$ *** $$
Let the set of functions Y = $\{ y_1(x),y_2(x),y_3(x) \} = \{x^2, \sin x, \cos x \}$, be linearly dependent. 
Since you did not mention the anything about where the function is coming from and going to and about the field.
I will suppose that, Y: $\mathbb{R}$ -> $\mathbb{R}$
Then by definition of linear dependence, there exists scalars (not all zero), such that, $$\ c_1.\mathbf{y_1(x)} + c_2.\mathbf{y_2(x)} + c_3.\mathbf{y_3(x)} = \mathbf{0(x)},\ \ \ \ x \in \mathbb{R},c_1,c_2,c_3 \in \mathbb{N} $$
$$ c_1.x^2 + c_2.\sin x + c_3.\cos x  = 0 ,\ \ \ \   x \in \mathbb{R},c_1,c_2,c_3 \in \mathbb{N}$$
$$ c_2.\sin x + c_3.\cos x  = -c_1.x^2 $$
$$ c_2.\sin x + c_3.\cos x  = c_4.x^2, \ \ \ \ c_4 = -c_1 $$
Is there a possibility that the manipulation of just the scalars $c_1, c_2, c_3$, we can equate the above equation?
Let $c_2$ = 0, then,
$$ c_3. \cos x = c_4.x^2 \ is \ only \ possible \ when \ c_3 = c_4 = 0$$
Let $c_3$ = 0, then,
$$ c_2. \sin x = c_4.x^2 \ is \ only \ possible \ when \ c_2 = c_4 = 0$$
Therefore, $ c_2.\sin x + c_3.\cos x  = c_4.x^2$ is only possible when $c_2 = c_3 = c_4 = 0$
Therefore, the set of function forms a linearly independent set.
$$ *** $$
Disclaimer: I don't have a definite reason why $ c_3. \cos x = c_4.x^2 \ is \ only \ possible \ when \ c_3 = c_4 = 0$, yet.
