1. The problem is to understand how Support Vector Machines (SVM) work and visualize their concept. 2. SVM is a supervised machine learning algorithm used for classification and regression tasks. It finds the best boundary (hyperplane) that separates classes in the feature space. 3. The key formula for the decision boundary is: $$\mathbf{w} \cdot \mathbf{x} + b = 0$$ where $\mathbf{w}$ is the weight vector perpendicular to the hyperplane, $\mathbf{x}$ is the input feature vector, and $b$ is the bias. 4. The support vectors are the data points closest to the hyperplane, which satisfy: $$\mathbf{w} \cdot \mathbf{x}_i + b = \pm 1$$ These points define the margin. 5. The margin width is given by: $$\text{Margin} = \frac{2}{\|\mathbf{w}\|}$$ Maximizing the margin leads to better generalization. 6. The optimization problem is: $$\min_{\mathbf{w},b} \frac{1}{2} \|\mathbf{w}\|^2$$ subject to $$y_i (\mathbf{w} \cdot \mathbf{x}_i + b) \geq 1$$ for all training points $(\mathbf{x}_i, y_i)$. 7. The diagram below shows two classes separated by the optimal hyperplane with support vectors on the margin boundaries.