Perceptron learning rule doesn't work i'm a little bit lost ... can you help me ?

So I have this table of date (each row give a point with its group)





So i took a random weight let's say : [1, -2]
H = 1 if n =&lt; 0
    0 otherwise

a= H([1,-2][6,3]) = H(0) = 1 but the target output is 0 ... so we have to update the weight:
w -> w - p = [5 , -5] .
Next : 

a= H([5,-5][3,3]= H(0)=1 the problem is : there the target output is 1 so we don't have to update the weight , but that's strange because the [5 -5] vector doesn't draw a linear separation between the 2 groups ...

Thanks for your help ? :)
I think you're mixing up the weight vector $w$ and the actual hyperplane defining the decision boundary. The way to interpret $w$ is as a vector which is orthogonal to any vector lying on the hyperplane. See these slides for some visualization.

As you've said, the decision is defined by $1$ if $w_1 x + w_2 y &gt; 0$ and $0$ otherwise. So to find the actual hyperplane (decision boundary) we can just solve for the place where the decision changes, which is at $0$. Using your $w$ we have 
$$
\begin{align*}
w_1 x + w_2 y &amp;= 0\\
\implies 5 x + -5y &amp;= 0\\
\implies x &amp;= y.
\end{align*}
$$
So the decision boundary is given by the line $y = x$. Plotting this with your points we see that this $w$ does indeed define a separating hyperplane.