Suppose I have an affine hyperplane of $\mathbb{R}^n$, say of the form $x + H$ for some vector $x$ and some hyperplane $H$. How would I go about describing this affine hyperplane by an equation $a_1x_1 + … + a_nx_n = b$? I am aware that I can describe the hyperplane $H$ by an equation $a_1x_1 + … + a_nx_n = 0$, but I'm not sure how to choose $b$ if I want to describe the affine hyperplane. Can this be done if I am only given some vector in the affine hyperplane (as well as the equation defining the hyperplane $H$)?
Describe an affine hyperplane by an equation
linear algebra
Best Answer
Let $H$ be a hyperplane defined by $$H = \{z \in \mathbb R^n: a_1z_1+\dots +a_nz_n=0\}$$ (i.e. each vector in $H$ is orthogonal to the vector $a=(a_1,\dots,a_n)$.
Now take any point $y$ in the affine hyperplane \begin{align*} x + H &= \{x + z \in \mathbb R^n: a_1z_1+\dots +a_nz_n=0\}\\ &= \{y \in \mathbb R^n: y-x \in H\}. \end{align*}
Then if you subtract $x$ from $y$, you end up in $H$. So $y-x$ must be orthogonal to $a$. That is, $$a_1(y_1-x_1)+\dots a_n(y_n-x_n)=0$$ which is equivalent to $$a_1y_1+\dots a_ny_1=a_1x_1+\dots +a_nx_n$$ Define $b=a_1x_1+\dots a_nx_1$ then $$x + H = \{z \in \mathbb R^n: a_1z_1+\dots +a_nz_n=b\}.$$