Sampled signals as a vector space

The sampled signal $\mathbf{x} = (\ldots, x_{-2}, x_{-1}, x_0, x_1, x_2, \ldots)$ is supposed to look like a vector, just as you learned in linear algebra. It just happens to be infinitely long. You can still treat it as you would regular vectors: add, subtract, pointwise multiply, scalar multiply, and take inner products. For instance, write $$\begin{eqnarray*} \mathbf{x} &=& (\ldots, 0, 0, 1, 2, 3, 0, 0, \ldots) \\ \mathbf{y} &=& (\ldots, 0, 0, 4, 5, 6, 0, 0, \ldots) \\ \mathbf{x} + \mathbf{y} &=& (\ldots, 0, 0, 5, 7, 9, 0, 0, \ldots) \\ \mathbf{y} - \mathbf{x} &=& (\ldots, 0, 0, 3, 3, 3, 0, 0, \ldots) \\ \mathbf{x} \cdot \mathbf{y} &=& (\ldots, 0, 0, 4, 10, 18, 0, 0, \ldots) \\ 10\mathbf{x} &=& (\ldots, 0, 0, 10, 20, 30, 0, 0, \ldots) \\ 10\mathbf{y} &=& (\ldots, 0, 0, 40, 50, 60, 0, 0, \ldots) \\ \langle \mathbf{x},\mathbf{y} \rangle &=& 1\cdot 4 + 2\cdot 5 + 3\cdot 6 = 32. \end{eqnarray*}$$

It is a nuisance to keep writing all these zeros in the infinite vectors, so sometimes we shorten things to condensed vectors, like $\mathbf{x} = (1,2,3)$ and $\mathbf{y} = (4,5,6)$. In this form, we have the 0-th entry in $\mathbf{x}$ as the first number that appears in the short vector. So $x_0 =1, x_1 = 2, x_2 = 3$, and the rest are zero. We do need to be careful when working with two or more vectors, that we continue to align the appropriate entries. So for instance $x_0$ must align with $y_0$, $x_1$ must align with $y_1$, and so on.