The Free Intercept

To avoid tracking a separate bias term $b$ , a common trick appends a constant $1$ to every feature vector $x$ , so the model becomes $w^\top x$ instead of $w^\top x + b$ . What does this do to the weight vector $w$ ?

It replaces $w$ with a fixed constant vector, since the extra feature is always $1$

It augments $w$ with one extra entry that absorbs the intercept $b$ , so the dot product alone reproduces $w^\top x + b$

It removes the need for any intercept, since the model becomes purely linear through the origin

It doubles the length of $w$ to account for both positive and negative bias