Linear function of first grade

Question

I have this function:

$f(\bar{x})=c_{0}+\sum_{1}^{n} c_{i}x_{i}$

with $\bar{x}\in \mathbb{R}^{n}$.

Is it correct to call this function : polynomial function of first grade ? And is it correct to call it hyperplane ?

Answer 1

1. Question: yes, $f$ is a polynomial of grade $1$.

2. Question: No, a hyperplane is a set and not a function.

Answer 2

Yes it is a polynomial function of degree one but it is not an hyperplane in $R^n$.

It could be an affine hyperplane in $R^{n+1}$

$$c_{0}+\sum_{1}^{n} c_{i}x_{i}=0$$