Kernel (linear algebra)

The kernel of a linear transformation is the subspace of its domain consisting of vectors mapped to the zero vector. It captures which inputs the transformation loses.

Null space: The set of solutions to a homogeneous system of linear equations, equivalently the kernel of its coefficient matrix. For a matrix transformation, its kernel is exactly the null space of the matrix.
Vector space: A set with vector addition and scalar multiplication satisfying the vector space axioms. The kernel is a subspace of the transformation’s vector-space domain.
System of linear equations: A collection of linear equations sharing the same unknowns. The solutions to a homogeneous system form a matrix kernel.
Cokernel: The quotient of a linear map’s codomain by its image. The kernel measures lost domain directions; the cokernel measures unattained codomain directions.
Rank–nullity theorem: A theorem stating that a linear map’s domain dimension equals its rank plus its nullity. It quantifies the tradeoff between outputs retained and dimensions collapsed into the kernel.
Linear transformation: A function between vector spaces that preserves vector addition and scalar multiplication. The kernel is defined for this structure-preserving kind of map.
Least squares: A method for estimating unknowns by minimizing the sum of squared residuals. A nontrivial kernel means some parameter changes leave a model’s predictions unchanged.
Image (linear algebra): The subspace of a linear transformation’s codomain consisting of its attainable outputs. Unlike the kernel, the image describes outputs rather than inputs sent to zero.
Injective function: A function is injective when distinct inputs always produce distinct outputs. A linear map is injective precisely when its kernel contains only the zero vector.
Zero vector: The additive identity in a vector space, unchanged by addition to any vector. The kernel collects precisely the inputs whose output is this vector.

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