One way of making linear regression applicable more widely is to use basis expansions, i.e., adding more features to the input set. Suppose that the data is described by a p-tuple, $(x_1 , x_2 , . . . , x_p )$. Comment on the utility of the following sets of features. Specifically describe the family of functions that can be represented by a linear combination of these features.

$(a)( x_1 , . . . , x_p , x_1^2 , x_1 x_2 , x_1 x_3 , . . . , x_1 x_p , x_ 2^2 , x_2 x_3 , . . . , x^2_p)$
$ (b) (x^2_1 , x^2_2 , . . . , x^2_p)$

How to solve this type of questions? Any hint or idea.

One way of making linear regression applicable more widely is to use basis expansions, i.e., adding more features to the input set. Suppose that the data is described by a p-tuple, $(x_1 , x_2 , . . . , x_p )$. Comment on the utility of the following sets of features. Specifically describe the family of functions that can be represented by a linear combination of these features.

$(a)( x_1 , . . . , x_p , x_1^2 , x_1 x_2 , x_1 x_3 , . . . , x_1 x_p , x_ 2^2 , x_2 x_3 , . . . , x^2_p)$
$ (b) (x^2_1 , x^2_2 , . . . , x^2_p)$

How to solve this type of questions? Any hint or idea.

My Attempt : I have studied linear regression from Stanford notes Now according to this the equation of the predicted value of $y$ is given by a linear equation of the feature variables, but here in this question the feature variables given are not linear so will it be of the same form?

Further by family of function does it imply finding the equation of $y$ or it has some other meaning? Here by $y$ I mean value being predicted using linear regression.