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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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  • The complex inner product is conjugate symmetric#23898
  • The composition of injective linear transformations is injective#24025
  • The composition of invertible linear transformations is invertible#24027
  • The composition of linear transformations is a linear transformation#24024
  • The composition of surjective linear transformations is surjective#24026
  • The conjugate of a matrix-scalar product is the product of the conjugates.#23468
  • The conjugate of a product of matrices is the product of the conjugates.#23469
  • The conjugate of a sum of vectors in C^n is the sum of the conjugates#23791
  • The conjugate of the sum of matrices is the sum of the conjugates.#23467
  • The conjugate of the transpose is the transpose of the conjugate.#23505
  • The conjugate of vector-scalar multiplication in C^n is the product of the conjugates.#23793
  • The coordinate vector relative to a given basis is a linear mapping to R^n (or C^n).#23992
  • The coordinate vector relative to a given basis is an injective linear mapping to R^n (or C^n).#23993
  • The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).#23994
  • The crazy vector space is a vector space.#24107
  • The determinant function exists.#23734
  • The determinant function is unique.#23735
  • The determinant of a block diagonal matrix is the product of the determinants of the blocks.#23731
  • The determinant of a matrix can be computed as a cofactor expansion across any row.#23719
  • The determinant of a matrix can be computed as a cofactor expansion down any column.#23720
  • The determinant of a matrix can be expressed as a product of the diagonal entries in a non-scaled echelon form.#23725
  • The determinant of a matrix measures the area/volume of the parallelogram/parallelipiped determined by its columns.#23746
  • The determinant of a triangular matrix is the product of the entries on the diagonal.#23744
  • The determinant of the inverse of A is the reciprocal of the determinant of A.#23730
  • The determinant of the matrix of a linear transformation is the factor by which the area/volume changes.#23747
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