Polynomial Regression and Features-Extension of Linear Regression

Priliminaries A Simple Linear Regression Least Squares Estimation Extending Linear Regression with Features1 The original linear regression is in the form:
[ begin{aligned} y(mathbf{x})&= b + mathbf{w}^T mathbf{x}\ &=w_01 + w_1x_1+ w_2x_2+cdots + w_{m+1}x_{m+1} end{aligned}tag{1} ]
where the input vector (mathbf{x}) and parameter (mathbf{w}) are (m)-dimension vectors whose first components are (1) and bias (w_0=b) respectively. This equation is linear for both the input vector and parameter vector. Then an idea come to us, if we set (x_i=phi_i(mathbf{x})) then equation (1) convert to: