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

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  • A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.#23824
  • A subset of a linearly independent set is linearly independent.#23980
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  • A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.#23934
  • A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace.#23923
  • Axioms of a vector space#23798
  • Bases#23828
  • Basic properties#23957
  • Basic properties of linear transformations#23863
  • Basic terminology#23412
  • Basic terminology and notation#23448
  • Block matrices#23567
  • Canonical forms of matrices#23608
  • Change of coordinates matrices are invertible#23998
  • Characteristic and minimal polynomials#23699
  • C^n is a vector space.#24098
  • Cofactors#23716
  • Composition#24022
  • Conjugating by a change of coordinates matrix converts matrix representations with respect to different bases.#23999
  • Conjugation#23464
  • Container for Linear Algebra#24123
  • Coordinates#23989
  • Coordinate vector spaces#23776
  • Cramer's rule#23722
  • Definition and terminology#23954
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  • Aspen Summer 2016 - Entanglement Matters#1213
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