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For linearly separable binary training data, a hard-margin SVM seeks a separating hyperplane with a large geometric margin. The formulation should be distinguished from a soft-margin SVM, which permits margin violations and uses a penalty controlled by C. A violation can include a point inside the margin even if its class is predicted correctly, so it is broader than misclassification alone. The kernel SVC decision function uses support vectors, training examples with nonzero dual coefficients. In a soft-margin model, those examples need not be only the nearest correctly classified points: margin-violating and misclassified examples can contribute. Kernel functions replace certain inner products in the formulation. They allow a decision function that is linear in an implicit feature space to be nonlinear in the original inputs. This does not prove that every possible dataset becomes perfectly separable or that kernel computation is always inexpensive. The radial basis function, or RBF kernel, measures similarity through squared input distance. Its gamma parameter affects the range of influence: larger gamma makes similarity fall more rapidly with distance, yielding a more localized influence. C controls the penalty assigned to violations relative to the regularization term, while the resulting boundary depends on data and both parameters. Scaling features can change distances and optimization geometry, making preprocessing important to assess. SVMs can be useful candidates in high-dimensional settings, including cases with more features than examples. That observation is not a guarantee of generalization, and kernel selection and regularization remain important.