I am not trying to cheat or anything, so any reference to online literature or MOOCs, that teach this stuff, will be highly appreciated.
The problem is to prove that the following optimization problem yields:
$w^* = \arg \min_{w} \tfrac{\gamma}{2} w^Tw+\frac{1}{2l}\sum_{i=1}^l{(x_i^T w-y_i)^2} $
$w^*= (X^TX + \gamma I)^{-1}X^Ty$
And that $X^TX+\gamma I$ is a positive definite matrix.
You can prove it just by setting the derivative to zero. You can find an example derivation in these slides
And $X^T X + \gamma I$ is always positive definite, because $X^T X$ is positive semi definite for any matrix $X$. Adding a scaled identity just increases each eigenvalue, hence the matrix is positive definite.