What is a linear kernel?

A linear kernel is the simplest kernel used by a Support Vector Machine (SVM): it computes the dot product of two input vectors. Unlike nonlinear kernels, it does not introduce a nonlinear feature mapping; a linear SVM learns a straight-line boundary in two dimensions or a hyperplane in higher dimensions.

Formula

K(x, z) = x · z = x1*z1 + x2*z2 + ... + xn*zn

Here x and z are two feature vectors of the same length. A linear kernel measures their unnormalized dot-product similarity. It is not Euclidean distance or cosine similarity: vector magnitudes affect its value.

Simple numerical example

Take two vectors x = (2, 3) and z = (4, 1):

K(x, z) = 2*4 + 3*1
        = 8 + 3
        = 11

Nothing has been transformed into extra polynomial or radial features. The calculation uses the original inputs directly.

How does it classify?

With a linear kernel, an SVM learns a decision score:

f(x) = w · x + b

• Positive score: one class.
• Negative score: the other class.
• Zero score: on the learned decision boundary.

Practical example: classify customer activity

Suppose two comparable-scale features summarize customer activity. We train on:

• (1, 1) and (1, 2) → Inactive (0).
• (4, 1) and (4, 2) → Active (1).

A maximum-margin linear boundary for this symmetric, separable dataset is x1 = 2.5:

• New customer (2, 1) → Inactive (0).
• New customer (3, 1) → Active (1).

The boundary is a straight line: this linear-kernel SVM cannot draw an arbitrary curved boundary on the supplied raw features.

Python example

PYTHON
from sklearn.svm import SVC

X = [[1, 1], [1, 2], [4, 1], [4, 2]]
y = [0, 0, 1, 1]

model = SVC(kernel='linear', C=100)
model.fit(X, y)

print(model.predict([[2, 1], [3, 1]]))  # [0 1]
print(model.coef_)                      # approximately [[0.667, 0.0]]
print(model.intercept_)                 # approximately [-1.667]

When should you use a linear kernel?

• Try it first when the classes are reasonably separable in the supplied feature space, or when you need a simpler, more interpretable boundary.
• It is often a strong choice for high-dimensional sparse features, such as bag-of-words text representations.
• For large datasets, consider `LinearSVC`: it uses a specialized linear solver and often scales better than SVC(kernel='linear'). Their implementations, default loss functions, and multiclass handling differ, so their fitted results need not be identical.
• Scale numerical features when appropriate, and select the regularization parameter C with validation data.

In short: a linear kernel computes K(x, z) = x · z and produces a linear SVM decision boundary. It is a practical baseline before considering RBF or polynomial kernels for nonlinear patterns.

References

• scikit-learn: SVM kernel functions
• scikit-learn: Linear kernel
• scikit-learn: LinearSVC API