Kernel (matrix)

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In linear algebra, the kernel or null space (also nullspace) of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of n-dimensional Euclidean space.[1] The dimension of the null space of A is called the nullity of A.

If viewed as a linear transformation, the null space of a matrix is precisely the kernel of the mapping (i.e. the set of vectors that map to zero). For this reason, the kernel of a linear transformation between abstract vector spaces is sometimes referred to as the null space of the transformation.

Contents

Definition

The kernel of an m × n matrix A is the set

where 0 denotes the zero vector with m components. The matrix equation Ax = 0 is equivalent to a homogeneous system of linear equations:

From this viewpoint, the null space of A is the same as the solution set to the homogeneous system.

Example

Consider the matrix

The null space of this matrix consists of all vectors (xyz) ∈ R3 for which

This can be written as a homogeneous system of linear equations involving x, y, and z:

This can be written in matrix form as:

Using Gauss-Jordan reduction, this reduces to:

Rewriting yields:

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