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

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A linear transformation is surjective if and only if the rank equals the dimension of the codomain.Close match
A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.Close match
A linear transformation is surjective if and only if the image of a basis is a spanning setClose match
Equivalence theorem for injective linear transformations: The rank of T is equals the number of columns in any matrix representation..Close match
Equivalence theorems for injective transformationsClose match
The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.Close match
A linear transformation is invertible if and only if it is injective and surjectiveClose match
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is onto/surjective.Close match
Equivalence theorem for injective linear transformations: The columns of the matrix of T are linearly independent.Close match
The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.Close match
Equivalence theorem for injective linear transformations: The rank of T is n.Close match
A linear transformation is injective on its generalized range space.Close match
Definition of onto/surjective linear transformationClose match
Definition of rank of a linear transformationClose match
Rank and nullityClose 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 one-to-one/injective linear transformationClose match
The rank of a matrix equals the rank of the linear transformation it represents.Close match
Definition of nullity of a linear transformationClose match

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