Why the weight vector is a linear combination of the inputs and the outputs in the Perceptron

I was studying Support Vector Machines and I've got stuck with this relation regarding the weight vector of the hyperplane.

$$w=\sum\limits_{i\in I}^{} y_i x_i$$

For reference, I'm studying from the Alex Smola slides and this relation is not justified.

That particular slide is about Perceptron algorithm, where initially $$w$$ is $$0$$ and you update it for each misclassified sample with the following update rule (there are slightly changed versions of this, but sticking with the slides): $$w\leftarrow w+y_ix_i$$ Because we start from $$0$$ and every update made is in terms of $$y_ix_i$$, the final version of the weights will be a linear combination of the input samples, i.e. something like $$w=\sum m_i y_i x_i$$
where $$m_i$$ is the number of times in the learning loop we updated for sample $$i$$. The concept of linear combination is correct, but the equation provided is wrong. If it were, we could have solved for $$w$$ directly, without the need of iterating until convergence: $$w=\sum y_i x_i=\sum_{y_i=1} x_i-\sum_{y_i=-1} x_i$$