Matrix Kernel (Null Space)

The kernel (also called the null space) of a matrix $ A $ of size $ m \times n $ is the set of vectors $ \vec{x} \in \mathbb{R}^n $ (or $ \mathbb{C}^n $) such that $ A \vec{x} = \vec{0} $, where $ \vec{0} $ is the zero vector in $ \mathbb{R}^m $. This set forms a vector subspace of $ \mathbb{R}^n $. It is exactly the null space of the linear transformation associated with $ A $, that is, the set of vectors mapped to the zero vector.

To compute the kernel of $ A $:

— Write the homogeneous system $ A\vec{x} = \vec{0} $.

— Reduce the matrix using Gaussian elimination until obtaining an echelon form (ideally reduced row echelon form).

— Identify pivot variables (columns with pivots) and free variables (columns without pivots).

— Express pivot variables in terms of free variables.

— Parameterize the solutions: the kernel is the set of all solutions, forming a vector subspace generated by vectors obtained by varying the parameters.

The kernel always contains at least the zero vector $ \vec{0} $, since $ A \vec{0} = \vec{0} $. Therefore, it is never empty.

If the kernel reduces to $ { \vec{0} } $, then the linear transformation associated with $ A $ is injective.

This means that distinct vectors always have distinct images.

The kernel is the set of vectors $ \vec{x} $ such that $ A \vec{x} = \vec{0} $. It is a vector subspace of $ \mathbb{R}^n $.

The image is the set of vectors $ \vec{y} $ such that there exists $ \vec{x} $ satisfying $ A\vec{x} = \vec{y} $. It is a vector subspace of $ \mathbb{R}^m $. The fundamental relation is: $ \dim(\operatorname{Ker}(A)) + \dim(\operatorname{Im}(A)) = n $, where $ n $ is the number of columns of $ A $ (rank theorem).

The rank of $ A $, denoted $ \operatorname{Rank}(A) $, measures the dimension of the image. The dimension of the kernel is called the nullity. These two quantities are related by:

$ \dim(\operatorname{Ker}(A)) + \operatorname{Rank}(A) = n $, where $ n $ is the number of columns of $ A $.

Thus, if $ A $ is invertible (so $ \operatorname{Rank}(A) = n $), then its kernel reduces to $ { \vec{0} } $.

If $ A $ is the zero matrix of size $ m \times n $, then $ A \vec{x} = \vec{0} $ for all $ \vec{x} \in \mathbb{R}^n $.

Therefore, $ \operatorname{Ker}(A) = \mathbb{R}^n $, which is the maximum possible kernel dimension for a matrix with $ n $ columns.