Assume you have two datasets of $($x$,$ y$)$:

$$ D$1=\{(1,+1),(0,1),(1,+1)\}.$

$$ D$2=\{(1,1),(0,+1),(1,1)\}.$

If you use feature $\backslash$phi $($x$)=$ x$,$ neither dataset can be linearly separated. You need to define a

two$-$dimensional feature $\backslash$phi $($x$)$ to fix this such that:

$$ A weight vector w$1$ can classify D$1$ perfectly $($i$.$e$.,$ w$1\backslash$phi $($x$)>0$ if x has label $+1$ and

w$1\backslash$phi $($x$)<0$ if x has label $1)$; and

$$ A weight vector w$2$ can classify D$2$ perfectly;

Note the two datasets share the same features $\backslash$phi $($x$)$ but the weight vectors can be different. $\backslash$phi $($x$)=$ x

normally can be re$-$written as $\backslash$phi $($x$)=[1,$ x$],$ which is regarded as one$-$dimensional feature.