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Extend a basis of the kernel to a basis of the finite-dimensional domain. The images of the added basis vectors form a basis of the range, giving nullity+rank=domain dimension.

Conditions: The linear map is defined on a finite-dimensional domain.; A basis for the kernel has been identified.

Extend a basis of the kernel to a basis of the finite-dimensional domain. The images of the added basis vectors form a basis of the range, giving nullity+rank=domain dimension.

Conditions: The linear map is defined on a finite-dimensional domain.; A basis for the kernel has been identified.

A linear map from a five-dimensional space to a three-dimensional space cannot be injective because its rank is at most three, so its nullity is at least two.

Conditions: The domain is five-dimensional.; The codomain is three-dimensional.; The map is linear.

A linear map from a five-dimensional space to a three-dimensional space cannot be injective because its rank is at most three, so its nullity is at least two.

Conditions: The domain is five-dimensional.; The codomain is three-dimensional.; The map is linear.

The kernel is spanned by the vector (1,1,0), and the range is all of the two-dimensional codomain. The nullity is one and the rank is two.

Conditions: The linear map is defined as T(x₁,x₂,x₃)=(x₁−x₂,2x₃).; The domain is three-dimensional and the codomain is two-dimensional.

The kernel is spanned by the vector (1,1,0), and the range is all of the two-dimensional codomain. The nullity is one and the rank is two.

Conditions: The linear map is defined as T(x₁,x₂,x₃)=(x₁−x₂,2x₃).; The domain is three-dimensional and the codomain is two-dimensional.