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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 columns of the matrix of T are linearly independent.

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Equivalence theorem for injective linear transformations: The image of a basis for V is a basis for the range of T.Close match
Equivalence theorem for injective linear transformations: The null space of T is 0.Close match
Equivalence theorems for injective transformationsClose match
Equivalence theorem for injective linear transformations: T(x)=0 only for x=0.Close match
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is one-to-one/injective.Close match
Equivalence theorem for injective linear transformations: The inverse of T is a linear transformation on its range.Close match
Equivalence theorem for nonsingular matrices: the columns of A are linearly independent.Close match
Equivalence theorem for nonsingular matrices: the rows of A are linearly independent.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 rank of T is equals the number of columns in any matrix representation..Close match
Equivalence theorem for injective linear transformations: The kernel of T is 0.Close match
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax has an inverse.Close match
The image of a linearly independent set under an injective linear transformation is linearly independent.Close match
Equivalence theorem for injective linear transformations: The nullity of T is 0.Close match
Equivalence theorem for injective linear transformations: The rank of T is n.Close match
A linear transformation is surjective if and only if the columns of its matrix span the codomain.Close match
A linear transformation is given by a matrix whose columns are the images of the standard basis vectorsClose match
Equivalent matrices represent the same linear transformation with resect to appropriate bases.Close match
Definition of one-to-one/injective linear transformationClose match
Eigenvalues agree and eigenvectors correspond through coordinatesClose match

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