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  • Definition of equivalent matrices
  • A matrix of rank k is equivalent to a matrix with 1 in the first k diagonal entries and 0 elsewhere.
  • Two matrices of the same size are equivalent if and only if they have the same rank.
  • Equivalent matrices represent the same linear transformation with resect to appropriate bases.
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  1. Two matrices of the same size are equivalent if and only if they have the same rank.

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  • Matrix equivalenceClose match
  • A matrix of rank k is equivalent to a matrix with 1 in the first k diagonal entries and 0 elsewhere.Close match
  • Definition of equivalent matricesClose match
  • Equivalence theorem for nonsingular matrices: the matrix A has rank n.Close match
  • The rank of a matrix equals the rank of the linear transformation it represents.Close match
  • Similarity of matrices is an equivalence relationClose match
  • Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.Close match
  • Equivalent matrices represent the same linear transformation with resect to appropriate bases.Close match
  • Definition of rank of a matrixClose match
  • Equivalence theorem for nonsingular matrices: the dimension of the column space of A is n.Close match
  • Rank–nullity theorem for an m-by-n matrixClose match
  • Definition of similar matricesClose match
  • Rank and nullityClose match
  • Similarity of matricesClose match
  • Equivalence theorem for nonsingular matrices: the matrix A has an inverse.Close match
  • Definition of equality of matricesClose 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 nonsingular matrices: the transpose of the matrix A has an inverse.Close match
  • Equivalence theorem for nonsingular matrices: the equation Ax=0 has only the trivial solution.Close match
  • Equivalence theorem for nonsingular matrices: the columns of A are a basis for R^n (or C^n).Close match

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