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In the book elements of statistical learning (e2 pg 130) it gives the definition of a hyperplane or affine set $L$ by the equation:

$$ f(x)=\beta_0+\beta^{\text{T}}x=0 $$

and lists some properties about hyperplanes / affine sets:

  1. For any two points $x_1$ and $x_2$ lying in $L$, $\beta^\text{T}(x_1-x_2)=0$ and hence $\beta^*=\beta/\lvert\lvert\beta\rvert\rvert$ is the vector normal to the surface of $L$

  2. For any point $x_0$ in $L$, $\beta^\text{T}x_0=-\beta_0$

  3. The signed distance of any point $x$ in $L$ is given by

$$ \begin{align} \beta^{*\text{T}}(x-x_0) &= \frac{1}{\lvert\lvert\beta\rvert\rvert}(\beta^\text{T}x+\beta_0)\\ &=\frac{1}{\lvert\lvert f'(x)\rvert\rvert}f(x) \end{align} $$

To my understanding the difference between an affine set and a hyperplane is that the latter passes through / contains the origin and the former is a translation from the origin.

Given the equation defining an affine set, it is clear to me that point 2 must hold. However, I do not see why one can derive that $\beta^*=\beta/\lvert\lvert\beta\rvert\rvert$ is the vector normal given $\beta^\text{T}(x_1-x_2)=0$. Nor do I understand how the norm of $\beta$ is the first derivative of $f$.

Can someone please elaborate?

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Let us refer to the figure below, from the book you mentioned. Let us define the hyperplane $L$ using: $$f(x)=\beta_0+\beta^Tx=0$$

If two points $x_1$ and $x_2$ lie in $L$, then they both satisfy the equation above, i.e.: $$\beta_0+\beta^Tx_1=0\\ \beta_0+\beta^Tx_2=0,$$ which gives you, if you subtract these two: $$\beta^T(x_1-x_2)=0.$$ You can interpret this as a dot product equal to zero indicating that $\beta$ is perpendicular to $(x_1-x_2)$ and since this latter is a vector that lies in $L$ this means that $\beta^*=\displaystyle\frac{\beta}{||\beta||}$ is normal to $L$.

Now, if $f(x)=\beta_0+\beta^Tx=0$, then the derivative $f'(x)=\beta$, and $||\beta|| = ||f'(x)||$.

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