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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 left null space of a matrix is a subspace of R^m (or C^m).#24111
  • The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A.#23768
  • The matrix representation of a composition of linear transformations is the product of the matrices.#24073
  • The matrix representation of a scalar multiple of linear transformations is the scalar multiple of the matrix.#24071
  • The matrix representation of a sum of linear transformations is the sum of the matrices.#24072
  • The matrix representation of the inverse of linear transformations is the inverse of the matricix.#24074
  • The minimal polynomial of a linear transformation exists and is unique.#24047
  • The minimal polynomial of a square matrix exists and is unique.#23704
  • The nonzero rows of an echelon form of a matrix are linearly independent.#23871
  • The nonzero rows of the reduced row-echelon form of a matris are a basis for the row space.#23879
  • The null space of a matrix is a subspace of R^n (or C^n).#24110
  • The null space of a matrix is the orthogonal complement of the column space.#23924
  • The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set.#23774
  • The number of pivots in the reduced row echelon form of a consistent system determines whether there is one or infinitely many solutions.#23773
  • The number of solutions to a linear system#23438
  • Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors#23820
  • Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0#23822
  • Theorem characterizing when a space is the direct sum of two subspaces#23976
  • Theorem describing matrix multiplication#23490
  • Theorem describing properties of the block matrices of the extended reduced row echelon form of a matrix#23551
  • Theorem describing spaces associated to the block matrices of the extended reduced row echelon form of a matrix#23552
  • Theorem describing the determinants of elementary matrices.#23745
  • Theorem describing the dimension of spaces associated to the block matrices of the extended reduced row echelon form of a matrix#23553
  • Theorem describing the vector form of sulutions to a linear system.#23444
  • The orthogonal complement of a subspace is a subspace.#23921
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