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Rank and nullity

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Definition of rank of a linear transformation
Definition of nullity of a linear transformation
A linear transformation is surjective if and only if the columns of its matrix span the codomain.

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Definition of nullity of a linear transformation

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Rank and nullityClose match
Equivalence theorem for injective linear transformations: The rank of T is n.Close match
Equivalence theorem for injective linear transformations: The nullity of T is 0.Close match
The rank plus the nullity of a linear transformation equals the dimension of the domain.Close match
Definition of rank of a linear transformationClose match
The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.Close match
Equivalence theorems for injective transformationsClose match
Definition of kernel/null space of linear transformationClose match
Definition of kernel of linear transformationClose match
Definition of nullity of a matrixClose match
Definition of nilpotent linear transformationClose match
Equivalence theorem for injective linear transformations: The null space of T is 0.Close match
Rank and nullityClose match
A linear transformation is surjective if and only if the rank equals the dimension of the codomain.Close match
Definition of generalized kernel/null space of linear transformationClose match
Definition of invertible/nonsingular linear transformationClose match
Finite-dimensional Fitting decomposition into generalized kernel and rangeClose match
Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.Close match
Definition of invertible linear transformationClose match
Rank–nullity theorem for an m-by-n matrixClose match

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