Introduction of New Notation for Multiple Features

Yesterday, someone mentioned some confusion over the notation h_θ(x) as the x was a single variable and now is a vector. I'll go through the general notation and then relate it back to the case where h_θ(x) = θ₀ + θ₁x.

The Hypothesis

When we just had one input variable x (size of house) and output y (cost of house) we had hypothesis

h_θ(x) = θ₀ + θ₁x.

This generalises to

h_θ(x_1, x_2, x_{3, ...,} x_n) = θ₀ + θ₁x₁ +  θ₂x₂ + θ₃x₃ + ... + θ_nx_n

This notation is rather cumbersome, so we use a more convenient vector form. In order to do this, we put in an addition x₀, which we set to one (as θ₀=θ₀*1=θ₀x₀.)

h_θ(x_0, x_1, x_2, x_{3, ...,} x_n) = θ₀x₀+ θ₁x₁ +  θ₂x₂ + θ₃x₃ + ... + θ_nx_n


Putting in this extra x₀ enables us to use the following notation.

I will use bold font to indicate vectors so x is the vector above. Using this notation, and writing θ as a row vector by taking its transpose θ^T =( θ₀, θ₁x₁, θ₂, θ₃x₃, ... , θ_n) we can rewrite the hypothesis as h_θ(x) = θ^Tx. Note that this x is the vector x.

Relating this to the Single Variable

For the single variable h_θ(x) = θ₀ + θ₁x, we rewrite it as
h(x_0, x₁) = θ₀ x₀+ θ₁x₁ = θ^Tx, that is,

h(x) = θ^Tx

where θ^T = (θ₀, θ₁) and

x=

(

x0 x1

).