Rank–nullity theorem

For a linear map between finite-dimensional vector spaces, the domain’s dimension equals the dimension of its image plus the dimension of its kernel.

Kernel (linear algebra): The subspace of a linear map’s domain consisting of vectors sent to zero. Its dimension is the part of the domain dimension lost under the map.
Rank–nullity theorem for matrices: The matrix form of rank–nullity: the number of columns equals matrix rank plus nullity. It gives a direct dimension count for the columns and solutions of a homogeneous system.
Linear map: A function between vector spaces that preserves vector addition and scalar multiplication. The theorem applies to linear maps between finite-dimensional spaces.
Rank–nullity theorem for infinite-dimensional spaces: A dimension formula for linear maps whose domain or codomain may be infinite-dimensional. Infinite dimensions require cardinal arithmetic, so the finite-dimensional statement’s intuition needs care.
Image (linear algebra): The subspace of a linear map’s codomain consisting of all outputs the map can produce. Its dimension is the map’s rank, the other term in the theorem.
System of linear equations: A collection of linear equations that must hold for the same unknowns. Nullity counts the free dimensions among solutions to the associated homogeneous system.
Vector space: A set with vector addition and scalar multiplication satisfying the vector-space axioms. The domain and codomain must be vector spaces for the theorem’s dimension statement.
Rank–nullity theorem for modules: An attempted rank-and-kernel dimension formula for module homomorphisms over rings. Modules over general rings need not have vector-space bases or dimensions obeying this formula.
Basis (linear algebra): A linearly independent set of vectors that spans a vector space. Extending a kernel basis supplies the vectors whose images form a basis of the image.
Invertible matrix: A square matrix that defines a linear map with an inverse. For a square matrix, zero nullity forces full rank and hence invertibility.

Linked from 22 pages