Kernel (linear algebra)

The kernel (also known as the null space) of a linear transformation or matrix is a fundamental concept in linear algebra. It represents the set of all vectors that are mapped to the zero vector by the transformation.

Formal Definition

Let ''T'': V → W be a linear transformation from a vector space V to a vector space W. The kernel of ''T'', denoted ker(''T'') or N(''T''), is defined as:

ker(''T'') = {v ∈ V | T(v) = 0_W}

where 0_W is the zero vector in W. Equivalently, for a matrix A, the kernel is the set of all vectors x such that Ax = 0, where 0 is the zero vector.

Properties

• The kernel of a linear transformation is always a subspace of the domain V. This means it contains the zero vector, is closed under addition, and is closed under scalar multiplication.
• The dimension of the kernel is called the nullity of the transformation or matrix. The nullity, along with the rank (dimension of the image), provides important information about the transformation via the rank-nullity theorem.
• The kernel can be used to determine whether a linear transformation is injective (one-to-one). A linear transformation is injective if and only if its kernel contains only the zero vector.
• Calculating the kernel often involves solving a system of homogeneous linear equations. Techniques like Gaussian elimination are commonly employed for this purpose.

Relationship to other concepts

The kernel is closely related to other important concepts in linear algebra, such as the image (or range), rank, and nullity. The rank-nullity theorem states that the dimension of the domain is equal to the sum of the rank and nullity of the linear transformation. Understanding the kernel is crucial for analyzing the properties of linear transformations and solving linear systems of equations. It plays a significant role in various applications, including solving differential equations and analyzing systems of linear equations.