All linear models make use of a "signal" $s$ which is a linear combination of the input vector ${\bf x}$ components weighed by the corresponding components in a weight vector ${\bf w}$.

$$ {\bf w} = \begin{bmatrix} w_0 & w_1 & ... & w_d \end{bmatrix}^T \\ s = w_0 + w_1 x_1 + \;...\; + w_d x_d = \sum_{i=0}^d w_i x_i = {\bf w} \cdot {\bf x} = {\bf w}^T {\bf x} $$

Note that the homogeneous representation (with the $1$ at the first component) allows us to include a constant offset using a more compact vector-only notation (instead of ${\bf w}^T {\bf x}+b$).