Linear Algebra: coordinates and basis I'm asked to show that if $f,g \in \mathrm{Sig}_3$ (sig stands for signal) have the coordinates $x,y \in \mathbb{R^4}$ with respect to the basis $(v_0, v_1, v_2, v_3)$, then 
$$\left<f,g\right>=\pi(x_1y_1+x_2y_2+x_3y_3+x_4y_4)$$
where
$$v_0=\frac{1}{\sqrt2}, 
v_1=\cos x,
v_2=\cos(2x),
v_3=\cos(3x)$$
and
$$C^0([0;2\pi])$$
Any idea how to show this? I'm completely stuck. All of the examples online use vectors with numbers, but I'm just given functions.

The question is a bit hard to understand because it's written in a weird order.  Usually the question goes (environment) (specifics) (statement to be proved).  If I understand your problem well, I think it would be better written this way:

Let $f$ and $g$ be elements of $\mathrm{Sig}_3$, which has inner product
  $$
    \left<f,g\right> = \int_0^{2\pi} f(x)g(x)\,dx
$$
  [I don't actually know what $\mathrm{Sig}_3$ is, even with the information that Sig stands for signal.  But I'm trying to work from the context.]  Let
  \begin{align*}
   v_0 &= \frac{1}{\sqrt{2}} \\
   v_1 &= \cos(x) \\
   v_2 &= \cos(2x) \\
   v_3 &= \cos(3x)
\end{align*}
  Suppose that with respect to the basis $(v_0, v_1, v_2, v_3)$, $f$ and $g$ have coordinates $x$ and $y$ in $\mathbb{R}^4$.  Show that
  $$
    \left<f,g\right> = \pi(x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)
$$

Please correct me if that's not what your problem is.  But notice how it starts with the environment of the vector space $\mathrm{Sig}_3$, along the definition of the inner product.  Then comes the specific information: for these two elements, we have these two coordinate vectors.  Finally, the statement we are to prove.
So I think you will be home free if you understand the meaning of this sentence:

Suppose that with respect to the basis $(v_0, v_1, v_2, v_3)$, $f$ and $g$ have coordinates $x$ and $y$ in $\mathbb{R}^4$.

What does this tell you about the formulas for $f(x)$ and $g(x)$?  Once you know this, you should be able to compute $\left<f,g\right>$ directly.
