$y = \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p$, unknowns $(x_1, \dots, x_p, y)$: Describes a hyperplane of affine space $\mathbb{R}^{p + 1}$?

Question

Let's say we have an equation $y = \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p$, where the unknowns are $(x_1, \dots, x_p,y)$. It is said that this describes a hyperplane $H$ of the affine space $\mathbb{R}^{p + 1}$. How is this a hyperplane of the affine space $\mathbb{R}^{p + 1}$? What is it about the mathematics of affine spaces that makes this so?

Answer 1

Affine:

Affine is a term to say it not a linear function, but shifted/translated( do not confuse with a nonlinear function either). Example $y = ax$ is linear as it includes the $0$ point. It(Linear) accepts the principle of superposition and homogeneity, whereas $y = ax + b$ is not a linear function but an affine function.

For your function the constant term $\beta_{0}$ makes it an affine function.

Next Hyperplane,

Hyperplane or a Hypersurface is a subspace of $n$ - dimensional space with one dimension less. Normally its something that splits an $n$ dimensional space.

General examples that are shown are.

"a point" is a hyperplane in 1-D space $\mathbb{R}^{1}$. It splits the line.

"a line" is a hyperplane in 2-D space $\mathbb{R}^{2}$. It splits the plane.

"a plane" is a hyperplane in 3-D space $\mathbb{R}^{3}$.

also an unit sphere is a "hypersurface" or a 2-D object in a 3-D space

Other way to see may be is , how many variables you require to define a point in a surface(in general)/plane. Always it is one dimension less.

So if a plane is defined as $Ax +By + Cy +d$ then $x,y,z$ is the space $\mathbb{R}^{3}$ plus a shift/translation/a constant term that is "real value" makes it to $\mathbb{R}^{3+1}$ which is an affine space(not $\mathbb{R}^{4}$, . So what above describes is a "hyperplane" a plane of one dimension less in that affine space.

So in general an affine space is defined by $\mathbb{R}^{n+1}$

If you don't have the constant term, then it is a plane containing the origin and you can relax mentioning it as an affine space.