I'm having a bit of confusion with a passage I'm reading about Maximal Margin Classifiers in the context of Support Vector Machines, which is making me think I need to go back to grade school. Here's the passage:
For some reason I can't seem to understand why this is the equation of a line. If I use the basic equation I know, $y=mx+b$, then rename $x$ to $X_{1}$, rename $y$ to $X_{2}$, rename $b$ to $\beta_{0}$, rename $m$ to $\beta_{1}$, then I get $X_{2}=\beta_{1}X_{1}+\beta_{0}$, or rearranged as: $\beta_{0}+\beta_{1}X_{1}-X_{2}=0$. Now flipping the sign I could maybe ignore as being unimportant, but why is there an extra coefficient term as well?
Best Answer
You could just multiply the equation $\beta_0+\beta_1X_1-X_2=0$ by another constant to give a "visible" coefficient on the $X_2$ as in equation (9.1).
More important, however, is that $y=mx+b$ is not the general equation of a line, because it will never give you a vertical line $x=c$. The form (9.1) is superior in this respect: it will give you vertical lines (when $\beta_2=0$), horizontal lines (when $\beta_1=0$), and everything in between.
(9.1) has the minor disadvantage that different equations will represent the same line, for example, $$1+2X_1+3X_2=0\quad\hbox{and}\quad 10+20X_1+30X_2=0\ ,$$ but as long as you are aware of this you will find that it rarely causes difficulties.