Data that can be separated by a line or hyperplane is know as linearly separable data. The hyperplane acts as a linear classifier.

A Support Vector Machine (SVM) is supervised machine learning algorithm used primarily for classification tasks. The core idea behind an SVM is to find the best boundary (or hyperplane in higher dimensions) that separates classes of data points with the maximum margin. The "margin" is defined as the distance between the separating hyperplane (decision boundary) and the closest data points from each class, which are called "support vectors." This concept can be extended to non-linear boundaries using something called the kernel trick.

Linear SVM Classification:

• Binary Classification: In its simplest form, when dealing with two classes, SVM finds a straight line (or hyperplane in higher dimensions) that separates the classes by the widest possible margin. This line is the decision boundary: data points on one side belong to one class, and those on the other side belong to another class.
• Support Vectors: The data points that determine the position and orientation of the hyperplane are the support vectors. These are the points closest to the hyperplane. The SVM essentially builds its decision-making model around these points.

import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import make_blobs
from sklearn.svm import SVC
from matplotlib import colors

# Create a synthetic dataset
X, y = make_blobs(n_samples=1000, centers=2, random_state=6)

# Initialize and train the SVM
svm = SVC(kernel='linear')
svm.fit(X, y)

# Plotting
fig, ax = plt.subplots()

# Plot the decision boundary
ax.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap=colors.ListedColormap(['blue', 'red']))
xlim = ax.get_xlim()
ylim = ax.get_ylim()

# Create grid to evaluate model
xx = np.linspace(xlim[0], xlim[1], 30)
yy = np.linspace(ylim[0], ylim[1], 30)
YY, XX = np.meshgrid(yy, xx)
xy = np.vstack([XX.ravel(), YY.ravel()]).T
Z = svm.decision_function(xy).reshape(XX.shape)

# Plot decision boundary and margins
ax.contour(XX, YY, Z, colors='k', levels=[-1, 0, 1], alpha=0.5,
           linestyles=['--', '-', '--'])

# Plot support vectors
ax.scatter(svm.support_vectors_[:, 0], svm.support_vectors_[:, 1], s=100,
           linewidth=1, facecolors='none', edgecolors='k')
plt.xlabel('Feature 1')
plt.ylabel('Feature 2')
plt.title('SVM Decision Boundary with Support Vectors')
plt.show()