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Equivalence theorems for injective transformations

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Definition of one-to-one/injective linear transformation
Equivalence theorem for injective linear transformations: The inverse of T is a linear transformation on its range.
Equivalence theorem for injective linear transformations: The null space of T is 0.
Equivalence theorem for injective linear transformations: The nullity of T is 0.
Equivalence theorem for injective linear transformations: The kernel of T is 0.
Equivalence theorem for injective linear transformations: T(x)=0 only for x=0.
Equivalence theorem for injective linear transformations: The rank of T is n.
Equivalence theorem for injective linear transformations: The rank of T is equals the number of columns in any matrix representation..
Equivalence theorem for injective linear transformations: The image of a basis for V is a basis for the range of T.
Equivalence theorem for injective linear transformations: The columns of the matrix of T are linearly independent.

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Equivalence theorem for injective linear transformations: The rank of T is equals the number of columns in any matrix representation..

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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
Equivalence theorems for injective transformationsClose match
A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.Close match
Equivalence theorem for injective linear transformations: The columns of the matrix of T are linearly independent.Close match
The rank of a matrix equals the rank of the linear transformation it represents.Close match
Equivalence theorem for injective linear transformations: The inverse of T is a linear transformation on its range.Close match
A linear transformation is surjective if and only if the columns of its matrix span the codomain.Close match
Equivalence theorem for injective linear transformations: The image of a basis for V is a basis for the range of T.Close match
Definition of rank of a linear transformationClose match
A linear transformation is surjective if and only if the rank equals the dimension of the codomain.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 nonsingular matrices: the linear transformation given by T(x)=Ax is an isomorphism.Close match
Equivalence theorem for injective linear transformations: The null space of T is 0.Close match
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is onto/surjective.Close match
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is one-to-one/injective.Close match
Two matrices of the same size are equivalent if and only if they have the same rank.Close match
Equivalence theorem for injective linear transformations: T(x)=0 only for x=0.Close match
Equivalence theorem for injective linear transformations: The kernel of T is 0.Close match
Rank and nullityClose match

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