Vector notation for weight and data

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Weight vector

The weight vector, is denoted as $w$, where:

$$w=\left[\begin{array}{c}{w}_1\\ w_2\\ \dots \\ w_n\end{array}\right]$$

Data(?) vector

The weight vector, is denoted as $x$, where:

$$x=\left[\begin{array}{c}{x}_1\\ x_2\\ \dots \\ x_n\end{array}\right]$$

Used to denote the position of one data point.

Implications

From Decision boundary (linear), we know that the decision boundary is notated as

$$w_1{x}_1+w_2{x}_2+\cdots +w_n{x}_n+b=0$$

As vectors

$$w\cdot x+b=0$$

Note: if treating $x$ and $w$ as matrices, need to transpose the first term, eg $w^T$,

$$\text{vector dot product of w and x}=-b\text{ (constant number)}$$

You may recognise this from Specialist Maths as the cartesian equation for a plane/hyperplane ($r\cdot n=r_0\cdot n$).

Here $r$, or the position vector, is represented by $x$. This makes $w$ the normal vector. $w$ will always be normal to classification (decision and margin) boundaries. $w$ affects the margin.