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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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  • Equivalence theorem for nonsingular matrices: the matrix A is a change-of-basis matrix.#23670
  • Equivalence theorem for nonsingular matrices: the matrix A represents the identity map with respect to some pair of bases.#23671
  • Equivalence theorem for nonsingular matrices: the matrix A row-reduces to the identity matrix.#23656
  • Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.#23668
  • Equivalence theorem for nonsingular matrices: the null space of the matrix A is {0}.#23667
  • Equivalence theorem for nonsingular matrices: there is a pivot position in every row of A.#23655
  • Equivalence theorem for nonsingular matrices: the rows of A are a basis for R^n (or C^n).#23652
  • Equivalence theorem for nonsingular matrices: the rows of A are linearly independent.#23650
  • Equivalence theorem for nonsingular matrices: the rows of A span R^n (or C^n).#23648
  • Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.#23660
  • Equivalence theorems for injective transformations#24077
  • Equivalent matrices represent the same linear transformation with resect to appropriate bases.#23605
  • Every basis for a vector space contains the same number of elements#24008
  • Every finite dimensional vector space over R (or C) is isomorphic to R^n (or C^n) for some n.#24006
  • Every matrix has an eigenvalue over the complex numbers.#23687
  • Every matrix is row-equivalent to a matrix in reduced row echelon form.#23548
  • Every matrix is row-equivalent to only one matrix in reduced row echelon form.#23549
  • Every nilpotent matrix is similar to one with 1 on subdiagonal blocks and all other entries 0.#23582
  • Every square matrix is conjugate#23566
  • Every square matrix is similar the sum of a diagonal and a nilpotent matrix.#23585
  • Every square matrix is similar to one in Jordan form.#23639
  • Example of a linear transformation on R^2: generic#23888
  • Example of a linear transformation on R^2: projection#23891
  • Example of a linear transformation on R^2: rotation#23890
  • Example of a linear transformation on R^2: shear#23889
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  • Aspen Summer 2016 - Entanglement Matters#1213
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